WEBVTT 00:00:18.675 --> 00:00:20.050 YUFEI ZHAO: We're about to embark 00:00:20.050 --> 00:00:22.900 on a new chapter in this course where 00:00:22.900 --> 00:00:25.520 I want to tell you about Szemeredi's graph regularity 00:00:25.520 --> 00:00:26.020 lemma. 00:00:46.820 --> 00:00:48.730 Szemeredi's graph regularity lemma 00:00:48.730 --> 00:00:52.400 is a very powerful tool in modern graph theory, developed 00:00:52.400 --> 00:00:54.440 back in the '70s. 00:00:54.440 --> 00:00:58.310 Today I want to show you the statement and the proof 00:00:58.310 --> 00:01:00.410 of this graph regularity lemma. 00:01:00.410 --> 00:01:05.120 And next time, we'll see how to apply the lemma for graph 00:01:05.120 --> 00:01:06.700 theoretic applications. 00:01:06.700 --> 00:01:09.050 And we'll also use it to give a proof of Roth's theorem. 00:01:11.850 --> 00:01:14.940 The idea of Szemeredi's regularity lemma 00:01:14.940 --> 00:01:24.460 is that if you are given a very large graph, G. 00:01:24.460 --> 00:01:28.090 And it's a fairly robust theorem, so 00:01:28.090 --> 00:01:33.370 any large, dense graph. 00:01:33.370 --> 00:01:38.640 And here, "dense" means, let's say, positive x density. 00:01:38.640 --> 00:01:50.050 Then it is possible to partition the vertex set of this graph G 00:01:50.050 --> 00:02:10.710 into a bounded number of pieces So that G looks random-like 00:02:10.710 --> 00:02:13.020 between most pairs of parts. 00:02:22.050 --> 00:02:24.930 So for instance, I might produce for you 00:02:24.930 --> 00:02:29.840 a partition of the vertex set into some number of parts. 00:02:29.840 --> 00:02:32.430 I'll draw five here. 00:02:32.430 --> 00:02:34.550 So you give me a graph G. I manage 00:02:34.550 --> 00:02:37.690 to produce for you this vertex partition 00:02:37.690 --> 00:02:42.130 so that if I look at between a typical pair of parts, 00:02:42.130 --> 00:02:46.870 you see here maybe the edge density is close to 0.2 00:02:46.870 --> 00:02:50.320 but otherwise, the bipartite graph looks 00:02:50.320 --> 00:02:53.580 like a random graph in some precise sense I will describe 00:02:53.580 --> 00:02:55.060 in a bit. 00:02:55.060 --> 00:02:57.150 And if you look at what the graph looks 00:02:57.150 --> 00:02:59.610 like between another pair of parts, 00:02:59.610 --> 00:03:01.770 maybe now it's a different x density. 00:03:01.770 --> 00:03:04.040 Maybe it's around 0.4. 00:03:04.040 --> 00:03:08.890 And again, looks like a random graph with that density. 00:03:08.890 --> 00:03:11.700 So in some sense, Szemeredi's regularity lemma 00:03:11.700 --> 00:03:14.760 is a universal structural description 00:03:14.760 --> 00:03:19.890 that allows you to approximate a graph by a bounded amount 00:03:19.890 --> 00:03:22.620 of information. 00:03:22.620 --> 00:03:24.300 So that's informally the idea. 00:03:24.300 --> 00:03:25.980 And you can already sense that this 00:03:25.980 --> 00:03:27.810 can be a very powerful tool. 00:03:27.810 --> 00:03:30.780 It doesn't matter what graph you input. 00:03:30.780 --> 00:03:36.630 You apply this lemma, and you get an approximate structural 00:03:36.630 --> 00:03:39.390 or as later on we'll see, it's also, in some sense, 00:03:39.390 --> 00:03:41.310 an analytic description of the graph. 00:03:43.840 --> 00:03:48.430 So the first part of today's lecture 00:03:48.430 --> 00:03:52.210 will develop just a statement of this regularity lemma. 00:03:52.210 --> 00:03:58.030 I'll show you what exactly do I mean by "random-like." 00:03:58.030 --> 00:04:03.294 Well, first let me give some definitions. 00:04:03.294 --> 00:04:06.350 I denote by the letter e if I input 00:04:06.350 --> 00:04:10.750 a pair of vertex sets, x and y. 00:04:13.900 --> 00:04:20.050 Here I might, later on, draw the subscript G 00:04:20.050 --> 00:04:23.530 if it's clear that I'm always talking about some graph 00:04:23.530 --> 00:04:27.220 G. So this is basically the number 00:04:27.220 --> 00:04:30.460 of edges between x and y. 00:04:41.710 --> 00:04:44.090 And I say "basically" because even though I 00:04:44.090 --> 00:04:47.720 will draw and depict everything as if x and y are disjoint 00:04:47.720 --> 00:04:51.320 sets, and that's the easiest case to think about, 00:04:51.320 --> 00:04:54.470 I'm also going to allow x and y to overlap, 00:04:54.470 --> 00:04:57.620 and also allow x and y to be the same set, in which case 00:04:57.620 --> 00:05:01.270 you should read a definition as to what this means. 00:05:01.270 --> 00:05:03.860 But it's fine to think of it as disjoint sets. 00:05:03.860 --> 00:05:07.760 So you're looking at a bipartite graph between x and y. 00:05:07.760 --> 00:05:13.100 We're also going to look at the edge density between x and y. 00:05:13.100 --> 00:05:17.300 And this is simply the number of edges divided 00:05:17.300 --> 00:05:23.570 by the product of the sizes of the sets, 00:05:23.570 --> 00:05:28.250 so what fraction of the possible pairs are actual edges. 00:05:28.250 --> 00:05:30.440 So from now on, I'll refer to this quantity 00:05:30.440 --> 00:05:31.720 as "edge density." 00:05:36.840 --> 00:05:41.370 So now, here's the definition of what "random-like" 00:05:41.370 --> 00:05:44.480 means for the purpose of Szemeredi's regularity lemma. 00:05:44.480 --> 00:05:50.880 So we define a notion of an epsilon regular pair 00:05:50.880 --> 00:05:52.140 to be as follows-- 00:05:52.140 --> 00:05:55.950 throughout, and later on, I will omit even saying this-- 00:05:55.950 --> 00:05:59.340 G will be some graph. 00:05:59.340 --> 00:06:05.220 And we're going to be looking at subsets of vertices of G. 00:06:05.220 --> 00:06:10.290 And we say that this pair of subsets of vertices 00:06:10.290 --> 00:06:18.250 is epsilon regular, again, in G, but later 00:06:18.250 --> 00:06:20.560 on I will even drop saying in G if it's clear 00:06:20.560 --> 00:06:22.710 which graph we're working with. 00:06:22.710 --> 00:06:25.540 So we say x and y is epsilon regular 00:06:25.540 --> 00:06:36.360 if for all subsets A of X, all B subsets of Y that are not 00:06:36.360 --> 00:06:48.720 too small, so each at least an epsilon proportion of the sets 00:06:48.720 --> 00:06:55.470 that they live in, we find that the x density between A and B 00:06:55.470 --> 00:06:59.160 differs from the x density between X and X 00:06:59.160 --> 00:07:03.010 by no more than epsilon. 00:07:03.010 --> 00:07:06.040 Let me draw you a picture. 00:07:06.040 --> 00:07:13.780 I have sets A and B. So I have sets X and Y in my graph G. 00:07:13.780 --> 00:07:21.000 And I want to say that the edges between X and Y are epsilon 00:07:21.000 --> 00:07:25.770 regular, so it's random-like, if the following holds-- 00:07:25.770 --> 00:07:31.700 that whenever I pick a subset A in the left set 00:07:31.700 --> 00:07:38.640 and a subset B of the right set, the edge density 00:07:38.640 --> 00:07:44.130 between A and B is approximately the same as the overall edge 00:07:44.130 --> 00:07:47.080 density between X and Y. 00:07:47.080 --> 00:07:51.100 So in particular, this bipartite graph, for instance, 00:07:51.100 --> 00:07:54.290 is not really dense in one part and really sparse 00:07:54.290 --> 00:07:55.150 in another part. 00:07:55.150 --> 00:07:58.390 Somehow the edges are evenly distributed 00:07:58.390 --> 00:08:00.810 in this precise manner. 00:08:00.810 --> 00:08:02.890 So that's the definition of epsilon regular. 00:08:02.890 --> 00:08:04.002 Yes, question. 00:08:04.002 --> 00:08:08.260 AUDIENCE: What is the epsilon for the size of A the same as 00:08:08.260 --> 00:08:11.345 epsilon for [INAUDIBLE]? 00:08:11.345 --> 00:08:12.720 YUFEI ZHAO: The question is here, 00:08:12.720 --> 00:08:15.763 why are we using the same epsilon here, here, and there? 00:08:15.763 --> 00:08:16.930 And that's a great question. 00:08:16.930 --> 00:08:19.070 So that's mostly out of convenience. 00:08:19.070 --> 00:08:21.780 So you could use different parameters. 00:08:21.780 --> 00:08:24.120 And they do play somewhat different roles, 00:08:24.120 --> 00:08:27.510 but at the end, we'll generally be looking 00:08:27.510 --> 00:08:29.740 at one type of epsilons. 00:08:29.740 --> 00:08:31.390 So we just make our life easier. 00:08:31.390 --> 00:08:33.539 So you could extend the definition 00:08:33.539 --> 00:08:37.600 by having an epsilon comma eta, if you like, 00:08:37.600 --> 00:08:40.020 but it will not be necessary for us, and mostly 00:08:40.020 --> 00:08:41.780 for simplification. 00:08:41.780 --> 00:08:43.144 Any more questions? 00:08:46.539 --> 00:08:47.509 All right. 00:08:51.890 --> 00:08:58.670 Now if you have a pair x, y that is not epsilon regular, 00:08:58.670 --> 00:09:00.690 I just want to introduce a piece of terminology. 00:09:00.690 --> 00:09:02.732 So you can read from the definition what it means 00:09:02.732 --> 00:09:04.160 to be not epsilon regular. 00:09:04.160 --> 00:09:06.370 And sometimes I will say "epsilon irregular," 00:09:06.370 --> 00:09:11.600 but to be precise, I'll stick with not epsilon regular. 00:09:16.680 --> 00:09:20.910 Then we can exhibit this A and B that witnesses 00:09:20.910 --> 00:09:23.080 the irregularity. 00:09:23.080 --> 00:09:29.700 So if x, y is not epsilon regular, then 00:09:29.700 --> 00:09:45.980 their irregularity as, we say it's "witnessed by" some pair 00:09:45.980 --> 00:09:52.380 A in X and B in Y, satisfying-- 00:09:52.380 --> 00:09:57.760 basically, you read the definition, 00:09:57.760 --> 00:10:01.200 and such that the density between A and B 00:10:01.200 --> 00:10:06.150 differs quite a bit from the density between X and Y. 00:10:06.150 --> 00:10:09.450 So when I say "to exhibit" or "to witness irregularity," 00:10:09.450 --> 00:10:11.640 that's what I mean. 00:10:11.640 --> 00:10:13.690 Now, there's a bit of an unfortunate nomenclature 00:10:13.690 --> 00:10:15.880 in graph theory, where previously, we 00:10:15.880 --> 00:10:19.960 said "irregular graphs" to mean that every vertex is degree D. 00:10:19.960 --> 00:10:23.710 And now we say "epsilon regular" to mean this. 00:10:23.710 --> 00:10:24.460 Sorry about that. 00:10:24.460 --> 00:10:27.070 These are both standard, so usually from context, 00:10:27.070 --> 00:10:28.900 it's clear which one is meant. 00:10:34.170 --> 00:10:37.050 So this is what it means for a single pair of vertex sets 00:10:37.050 --> 00:10:38.670 to be epsilon regular. 00:10:38.670 --> 00:10:40.680 But now I give you a graph. 00:10:40.680 --> 00:10:44.110 And I give you a partition of the vertex set. 00:10:44.110 --> 00:10:45.900 So what does it mean for that partition 00:10:45.900 --> 00:10:47.955 to be epsilon regular? 00:10:47.955 --> 00:10:49.818 And here's the second definition. 00:10:53.940 --> 00:11:05.980 So an epsilon regular partition, we say that a partition-- 00:11:05.980 --> 00:11:08.120 and generally, I will denote partition 00:11:08.120 --> 00:11:12.860 by curly letters such as that, P. So the partition 00:11:12.860 --> 00:11:23.030 will divide a vertex set into a bunch of subsets. 00:11:23.030 --> 00:11:30.880 So we say that that partition is epsilon regular 00:11:30.880 --> 00:11:33.430 if the following is true-- 00:11:33.430 --> 00:11:45.590 if I sum over all pairs of irregular 00:11:45.590 --> 00:11:55.100 or pairs of vertex sets that are not epsilon regular, 00:11:55.100 --> 00:11:59.370 so over Vi, Vj not epsilon regular, and sum 00:11:59.370 --> 00:12:07.020 over the product of their sizes, then what I would like 00:12:07.020 --> 00:12:12.540 is for the sum to be at most epsilon times 00:12:12.540 --> 00:12:18.320 the number of pairs of vertices in G. In other words, 00:12:18.320 --> 00:12:22.060 a small fraction of pairs of vertices, 00:12:22.060 --> 00:12:24.790 not necessarily edges, but just pairs of vertices, 00:12:24.790 --> 00:12:30.050 lie between pairs of vertex parts 00:12:30.050 --> 00:12:33.060 that are not epsilon regular. 00:12:33.060 --> 00:12:37.350 So for instance, if you do not have epsilon-- 00:12:37.350 --> 00:12:39.360 if all of your pairs are epsilon regular, 00:12:39.360 --> 00:12:41.850 then the partition is epsilon regular. 00:12:41.850 --> 00:12:46.610 But I do allow a small number of blemishes. 00:12:46.610 --> 00:12:48.003 And that will be necessary. 00:12:53.680 --> 00:12:56.340 Just to clarify a subtle point here, 00:12:56.340 --> 00:13:03.310 here I do allow in the summation i equals to j, 00:13:03.310 --> 00:13:05.990 although in practice it doesn't really matter. 00:13:05.990 --> 00:13:08.690 You'll see that it's not really going to come up as an issue. 00:13:14.680 --> 00:13:16.430 And one of the reasons that it's not 00:13:16.430 --> 00:13:19.280 going to come up as an issue is usually 00:13:19.280 --> 00:13:22.220 when we apply this lemma, we're going to have a lot of parts. 00:13:22.220 --> 00:13:24.330 In fact, we can make sure that there 00:13:24.330 --> 00:13:26.700 is a minimum number of parts. 00:13:26.700 --> 00:13:34.010 And if none of the parts are too big, then having i equals to j 00:13:34.010 --> 00:13:36.030 contributes very little to that sum anyway. 00:13:41.410 --> 00:13:49.530 In particular, if all the set sizes 00:13:49.530 --> 00:13:51.870 in this partition are roughly the same-- 00:13:55.730 --> 00:14:01.950 so if they're all roughly 1 over k fraction of the entire vertex 00:14:01.950 --> 00:14:02.970 set-- 00:14:02.970 --> 00:14:13.410 then that statement up there being epsilon regular partition 00:14:13.410 --> 00:14:16.500 up to changing this epsilon is basically the same 00:14:16.500 --> 00:14:23.030 as saying that fewer than epsilon fraction of the pairs 00:14:23.030 --> 00:14:29.670 Vi, Vj are not epsilon regular. 00:14:29.670 --> 00:14:32.840 And here, if k is large enough, I 00:14:32.840 --> 00:14:35.910 can even let you make i and j different. 00:14:35.910 --> 00:14:39.110 It's not going to affect things after small changes in epsilon. 00:14:42.450 --> 00:14:45.140 So when it comes to-- so for people who are seeing 00:14:45.140 --> 00:14:48.040 Szemeredi's regularity lemma for the first time-- 00:14:48.040 --> 00:14:51.370 I think that's maybe all of you, or most of you-- 00:14:51.370 --> 00:14:54.820 I don't want you to focus on the precise statements 00:14:54.820 --> 00:14:56.710 so much as the spirit of the lemma. 00:14:56.710 --> 00:14:59.220 Because if you get too nitty gritty with 00:14:59.220 --> 00:15:02.350 is that the same as that epsilon, you get very confused 00:15:02.350 --> 00:15:03.860 very quickly. 00:15:03.860 --> 00:15:06.820 So I want you to focus on the spirit of this lemma. 00:15:06.820 --> 00:15:09.190 I will state everything precisely, 00:15:09.190 --> 00:15:13.588 but the idea is that most pairs are not epsilon regular. 00:15:13.588 --> 00:15:15.630 And don't worry too much about if you are allowed 00:15:15.630 --> 00:15:17.110 to take i equals to j or not. 00:15:19.960 --> 00:15:22.490 So now we're ready to state Szemeredi's regularity lemma. 00:15:38.450 --> 00:15:42.530 And it says that for every epsilon, 00:15:42.530 --> 00:15:50.170 there exists some constant M depending only on epsilon such 00:15:50.170 --> 00:16:05.520 that every graph has an epsilon regular partition 00:16:05.520 --> 00:16:09.140 into at most M parts. 00:16:14.330 --> 00:16:17.390 You give me the epsilon, for example 1%, 00:16:17.390 --> 00:16:20.240 and there exists some constant such 00:16:20.240 --> 00:16:25.280 that every graph has a 1% regular partition 00:16:25.280 --> 00:16:28.838 into a bounded number of parts. 00:16:28.838 --> 00:16:30.630 In particular-- and this is very important, 00:16:30.630 --> 00:16:32.130 make sure you understand this part-- 00:16:32.130 --> 00:16:35.840 that the number of parts does not depend 00:16:35.840 --> 00:16:36.890 on the size of the graph. 00:16:48.200 --> 00:16:50.700 Now, it's true that for some graphs, maybe 00:16:50.700 --> 00:16:53.600 you do need very many parts. 00:16:53.600 --> 00:16:58.190 But the number of parts does not get substantially bigger, 00:16:58.190 --> 00:17:02.480 or does not exceed this bound, even if you look at graphs 00:17:02.480 --> 00:17:04.630 that have unbounded size. 00:17:04.630 --> 00:17:07.310 So it is really a universal theorem in the sense 00:17:07.310 --> 00:17:10.290 that it's independent of the size of the graph. 00:17:13.839 --> 00:17:16.200 Any questions about the statement of this theorem? 00:17:16.200 --> 00:17:16.876 Yes. 00:17:16.876 --> 00:17:19.209 AUDIENCE: So in the informal statement at the beginning, 00:17:19.209 --> 00:17:20.759 you said G was a large, dense graph. 00:17:20.759 --> 00:17:21.801 YUFEI ZHAO: That's right. 00:17:21.801 --> 00:17:23.859 AUDIENCE: Is the dense condition appropriate anywhere in here? 00:17:23.859 --> 00:17:26.220 YUFEI ZHAO: So the question is, why did I say 00:17:26.220 --> 00:17:29.220 that G is a large, dense graph? 00:17:29.220 --> 00:17:30.900 And that's a great question. 00:17:30.900 --> 00:17:35.370 And that's because if G had a sub-linear number of edges, 00:17:35.370 --> 00:17:39.270 then I claim that all-- 00:17:39.270 --> 00:17:46.090 if you look at the definition of epsilon regular pair, 00:17:46.090 --> 00:17:52.180 and your epsilon is a constant, and if your edge densities 00:17:52.180 --> 00:17:54.250 are sub-linear, then all of these guys, 00:17:54.250 --> 00:17:56.170 they are little o of 1. 00:17:56.170 --> 00:17:57.520 They go to 0. 00:17:57.520 --> 00:18:03.170 So trivially, you will satisfy the epsilon regular condition. 00:18:03.170 --> 00:18:04.940 So if your graph is sparse-- 00:18:04.940 --> 00:18:09.410 sparse in the sense of having sub-quadratic number of edges-- 00:18:09.410 --> 00:18:15.070 then you trivially obtain epsilon regularity. 00:18:15.070 --> 00:18:17.620 And so the theorem is still true. 00:18:17.620 --> 00:18:19.000 It's just not meaningful. 00:18:19.000 --> 00:18:20.050 It's just not useful. 00:18:20.050 --> 00:18:22.720 But there are settings where having sparse graphs-- 00:18:22.720 --> 00:18:25.410 and we'll come back to this later in the course-- 00:18:25.410 --> 00:18:27.757 it's important to explore what happens to sparse graphs. 00:18:27.757 --> 00:18:28.257 Yeah. 00:18:28.257 --> 00:18:31.670 AUDIENCE: So that M is independent of G. 00:18:31.670 --> 00:18:35.030 YUFEI ZHAO: Yes, M is independent of G. M depends 00:18:35.030 --> 00:18:35.900 only on epsilon. 00:18:39.108 --> 00:18:41.566 AUDIENCE: M is really large, but there's no enough vertices 00:18:41.566 --> 00:18:42.290 in the graph. 00:18:42.290 --> 00:18:42.990 YUFEI ZHAO: OK, question is, what 00:18:42.990 --> 00:18:44.948 happens when M is very large, but there are not 00:18:44.948 --> 00:18:47.170 enough vertices in the graph? 00:18:47.170 --> 00:18:50.620 Well, if your M is a million, and your graph only 00:18:50.620 --> 00:18:52.690 has 1,000 vertices, what you can do 00:18:52.690 --> 00:18:54.610 is have every vertex be its own part. 00:18:58.730 --> 00:19:02.060 Every vertex is its own part, a singleton partition. 00:19:02.060 --> 00:19:05.240 And you can check that that partition satisfies 00:19:05.240 --> 00:19:06.980 the properties. 00:19:06.980 --> 00:19:12.152 Every pair is a single edge and it's epsilon regular. 00:19:12.152 --> 00:19:13.090 Yeah. 00:19:13.090 --> 00:19:15.163 AUDIENCE: So in the definition, is it 00:19:15.163 --> 00:19:16.330 sort of like all or nothing? 00:19:16.330 --> 00:19:18.788 You can either [INAUDIBLE] epsilon regularity [INAUDIBLE].. 00:19:18.788 --> 00:19:21.110 Do you get anything where if you, like, say, 00:19:21.110 --> 00:19:24.838 make this more continuous, so you allow for it to be-- 00:19:24.838 --> 00:19:26.974 you quantify how irregular it is, and then can 00:19:26.974 --> 00:19:28.790 you make [INAUDIBLE]? 00:19:28.790 --> 00:19:30.660 YUFEI ZHAO: OK, so my understanding what 00:19:30.660 --> 00:19:33.360 you're asking is in the definition up there, 00:19:33.360 --> 00:19:36.300 the sum is-- 00:19:36.300 --> 00:19:39.360 we put the pair in the sum of this epsilon regular, 00:19:39.360 --> 00:19:40.680 and otherwise don't put it. 00:19:40.680 --> 00:19:44.550 Is there some gradual way to put some measure of irregularity 00:19:44.550 --> 00:19:45.450 into that sum? 00:19:45.450 --> 00:19:48.410 And there are versions of regularity lemma that do that, 00:19:48.410 --> 00:19:51.030 but they are all, in spirit, morally 00:19:51.030 --> 00:19:53.885 the same as that one there. 00:19:53.885 --> 00:19:54.385 Yeah. 00:19:54.385 --> 00:19:55.927 AUDIENCE: In the informal definition, 00:19:55.927 --> 00:19:57.200 what does "random-like" mean? 00:19:57.200 --> 00:19:58.950 YUFEI ZHAO: So in the informal definition, 00:19:58.950 --> 00:20:00.510 what does "random-like" mean? 00:20:00.510 --> 00:20:03.330 This is the formal definition of what "random-like" means. 00:20:03.330 --> 00:20:06.240 So actually later on in the course, one of the chapters 00:20:06.240 --> 00:20:09.390 will explore what pseudo-random graphs are. 00:20:09.390 --> 00:20:10.980 So pseudo-random graph, in some sense, 00:20:10.980 --> 00:20:12.480 means graphs that are not random, 00:20:12.480 --> 00:20:15.030 but behave in some sense like random. 00:20:15.030 --> 00:20:19.550 So "random-like" generally just means that in some aspect, 00:20:19.550 --> 00:20:23.970 in some property, it looks like a random object. 00:20:23.970 --> 00:20:27.830 And this is one way that something can look like random. 00:20:27.830 --> 00:20:31.910 So a random graph has this property, 00:20:31.910 --> 00:20:34.580 but random graphs also have many other properties 00:20:34.580 --> 00:20:37.790 that are not being exhibited in this definition. 00:20:37.790 --> 00:20:42.260 But this is one way that graph can look like random. 00:20:42.260 --> 00:20:43.430 So that's a great question. 00:20:43.430 --> 00:20:45.680 And we'll come back to that topic later in the course. 00:20:47.655 --> 00:20:49.030 All of these are great questions. 00:20:49.030 --> 00:20:51.870 So Szemeredi's regularity lemma, the first time you see it, 00:20:51.870 --> 00:20:54.310 it can look somewhat scary. 00:20:54.310 --> 00:20:55.910 But I want you to try to understand 00:20:55.910 --> 00:20:56.900 it more conceptually. 00:20:56.900 --> 00:20:58.550 So please do ask questions. 00:21:05.900 --> 00:21:09.350 Before diving into the proof, I want to make a few more remarks 00:21:09.350 --> 00:21:10.709 about a statement. 00:21:15.300 --> 00:21:18.130 It is possible to-- 00:21:18.130 --> 00:21:21.840 we will prove this version of the regularity lemma. 00:21:21.840 --> 00:21:25.930 But as I mentioned, it is the spirit of the regularity lemma 00:21:25.930 --> 00:21:27.360 that I care more about. 00:21:27.360 --> 00:21:29.610 And it's a very robust statement. 00:21:29.610 --> 00:21:32.697 You can add on extra declarations that somehow 00:21:32.697 --> 00:21:33.780 doesn't change the spirit. 00:21:33.780 --> 00:21:35.613 And the proof will be more or less the same, 00:21:35.613 --> 00:21:38.770 but for various applications will be slightly more useful. 00:21:38.770 --> 00:21:41.300 So in particular, it is possible to make 00:21:41.300 --> 00:21:43.329 the partition equitable. 00:21:56.060 --> 00:21:58.000 And "equitable partition" sometimes 00:21:58.000 --> 00:22:04.540 is also called an "equipartition," meaning 00:22:04.540 --> 00:22:09.490 that it has such that all the Ai's, all 00:22:09.490 --> 00:22:13.240 the Bi's have sizes differing by at most 1. 00:22:19.090 --> 00:22:22.910 So basically, all the parts have the same size up to at most 1, 00:22:22.910 --> 00:22:25.700 because of divisibility. 00:22:25.700 --> 00:22:29.570 So let me state a version of regularity lemma 00:22:29.570 --> 00:22:31.690 for equitable partitions. 00:22:31.690 --> 00:22:37.010 So for every epsilon in m, little m0, 00:22:37.010 --> 00:22:48.560 there exists a big M such that every graph 00:22:48.560 --> 00:23:05.250 has an epsilon regular equitable partition of the vertex set 00:23:05.250 --> 00:23:13.440 into k parts, where k is at least little m, so I can 00:23:13.440 --> 00:23:15.420 guarantee a minimum number of parts, 00:23:15.420 --> 00:23:17.940 and at most some bounded number. 00:23:17.940 --> 00:23:24.750 Again this bound may depend on your inputs epsilon and m0, 00:23:24.750 --> 00:23:28.740 but it does not depend on the graph itself. 00:23:28.740 --> 00:23:30.950 And you see the slightly stronger conclusion 00:23:30.950 --> 00:23:33.170 for many applications is more convenient, 00:23:33.170 --> 00:23:37.120 to use this formulation. 00:23:37.120 --> 00:23:41.890 And I will comment on how you may modify the proof that we'll 00:23:41.890 --> 00:23:46.820 see today into one where you can guarantee equitability. 00:23:46.820 --> 00:23:57.750 And you see that for this m, little m0 too small, 00:23:57.750 --> 00:24:02.040 for example, if it's somewhat larger than 1 over epsilon, 00:24:02.040 --> 00:24:06.260 when you look at the definition of epsilon regular partition, 00:24:06.260 --> 00:24:10.080 it suffices to check that at most epsilon k squared, 00:24:10.080 --> 00:24:16.225 epsilon fraction of the pairs, Vi, Vj 00:24:16.225 --> 00:24:20.710 is epsilon regular over i different 00:24:20.710 --> 00:24:23.950 from j, again up to changing epsilon, 00:24:23.950 --> 00:24:25.580 let's say, by a factor of 2. 00:24:25.580 --> 00:24:28.540 So all of these definitions are basically 00:24:28.540 --> 00:24:30.940 the same up to small changes in the parameters. 00:24:42.420 --> 00:24:45.080 Next time, we'll see how to apply the regularity lemma. 00:24:45.080 --> 00:24:47.960 And we will apply it in the first form, 00:24:47.960 --> 00:24:49.682 but you see the second form guarantees 00:24:49.682 --> 00:24:51.140 you a somewhat stronger conclusion, 00:24:51.140 --> 00:24:53.370 and sometimes more convenient to use. 00:24:53.370 --> 00:24:55.430 So for example on the homework problems, 00:24:55.430 --> 00:24:59.900 if you wish to use the second form, then please go ahead. 00:24:59.900 --> 00:25:01.760 Just make your life somewhat easier, 00:25:01.760 --> 00:25:04.040 but it essentially captures all the spirit 00:25:04.040 --> 00:25:05.720 of Szemeredi's regularity. 00:25:09.815 --> 00:25:10.690 Any questions so far? 00:25:13.610 --> 00:25:17.600 I want to explain the idea of the proof of the regularity 00:25:17.600 --> 00:25:19.220 lemma. 00:25:19.220 --> 00:25:21.980 And this is a very important technique 00:25:21.980 --> 00:25:25.493 in this area called the "energy increment argument." 00:25:35.850 --> 00:25:38.110 Here's the idea. 00:25:38.110 --> 00:25:43.710 We start with some partition, so for example, 00:25:43.710 --> 00:25:46.250 the trivial partition-- 00:25:46.250 --> 00:25:49.680 and by that I mean you only have one part. 00:25:52.295 --> 00:25:53.670 All the vertices are in one part. 00:25:53.670 --> 00:25:55.830 You're not doing anything to the vertex set. 00:25:55.830 --> 00:25:58.770 It's one gigantic part. 00:25:58.770 --> 00:26:01.350 Or if you're looking at some other variant, 00:26:01.350 --> 00:26:03.120 you can easily modify the proof. 00:26:03.120 --> 00:26:09.050 So for example, you can also look at an arbitrary partition 00:26:09.050 --> 00:26:13.523 into little m0 parts, if you wish to have 00:26:13.523 --> 00:26:14.690 that as your starting point. 00:26:14.690 --> 00:26:16.898 So or I'm saying is that this proof is fairly robust. 00:26:19.880 --> 00:26:23.050 And we're going to do some iterations. 00:26:26.680 --> 00:26:37.970 So as long as your partition is not epsilon regular, 00:26:37.970 --> 00:26:44.660 we will do something to the partition to move forward. 00:26:44.660 --> 00:26:54.590 And what we will do is look at each pair 00:26:54.590 --> 00:26:59.600 of parts in your partition that's not epsilon regular. 00:27:04.170 --> 00:27:05.960 Well, if they're not epsilon regular, 00:27:05.960 --> 00:27:08.390 then I can find a pair of subsets 00:27:08.390 --> 00:27:17.520 which are denoted by the A's that witnesses 00:27:17.520 --> 00:27:26.880 this non regularity, that witnesses the irregularity. 00:27:26.880 --> 00:27:29.370 And we start with some partition. 00:27:29.370 --> 00:27:33.210 So now let us refine the partition 00:27:33.210 --> 00:27:36.720 into a partition in even more parts 00:27:36.720 --> 00:27:53.090 by simultaneously refining the partition using 00:27:53.090 --> 00:28:00.360 all of these Ai, j's that we found in the step above. 00:28:00.360 --> 00:28:01.930 So you start with some partition. 00:28:01.930 --> 00:28:05.560 If it is not regular, I can chop up 00:28:05.560 --> 00:28:08.120 the various parts in some way. 00:28:08.120 --> 00:28:12.090 So I start with some partition over here. 00:28:12.090 --> 00:28:15.930 And what we are going to do is, let's say between these two, 00:28:15.930 --> 00:28:18.180 it's not epsilon regular, so I can 00:28:18.180 --> 00:28:23.260 find some pairs of vertex sets that exhibits the irregularity. 00:28:23.260 --> 00:28:24.650 I chop it up. 00:28:24.650 --> 00:28:31.230 And I can keep further chopping up the rest of the parts. 00:28:31.230 --> 00:28:33.420 If these two parts are not epsilon regular, 00:28:33.420 --> 00:28:35.580 then I chop it up like that. 00:28:35.580 --> 00:28:38.802 And I can keep on doing it. 00:28:38.802 --> 00:28:40.260 And originally, I have three parts. 00:28:40.260 --> 00:28:43.040 Now I have 12 parts. 00:28:43.040 --> 00:28:45.580 And this is a refined partition. 00:28:45.580 --> 00:28:49.130 And now I repeat until I am done. 00:28:49.130 --> 00:28:52.490 I am done when I obtain a partition that 00:28:52.490 --> 00:28:55.750 is epsilon regular. 00:28:55.750 --> 00:28:58.260 Now, the basic question when it comes to the strategy 00:28:58.260 --> 00:29:00.270 is, are you ever going to be done? 00:29:00.270 --> 00:29:02.570 When are you going to be done? 00:29:02.570 --> 00:29:04.650 And if this process goes on forever 00:29:04.650 --> 00:29:07.047 or goes on for a very long time, then you 00:29:07.047 --> 00:29:08.130 might have a lot of parts. 00:29:08.130 --> 00:29:09.600 But we want to guarantee that there 00:29:09.600 --> 00:29:12.680 is a bounded number of parts. 00:29:12.680 --> 00:29:18.470 So what we will show is that-- 00:29:18.470 --> 00:29:22.360 to show that you have a small number of parts, 00:29:22.360 --> 00:29:24.740 in other words, why does this process even stop-- 00:29:30.915 --> 00:29:33.080 and in particular, we want it to stop 00:29:33.080 --> 00:29:38.320 after a small number of steps, after a bounded number 00:29:38.320 --> 00:29:40.750 of steps. 00:29:40.750 --> 00:29:43.560 And to do this, we will define some notion called 00:29:43.560 --> 00:29:47.627 an "energy" of a partition. 00:29:51.370 --> 00:29:55.280 And this energy will increase. 00:29:55.280 --> 00:29:58.300 So first of all, the energy is some quantity 00:29:58.300 --> 00:30:02.980 that we'll define that lies between 0 and 1. 00:30:02.980 --> 00:30:05.860 It's some real number lying between 0 and 1. 00:30:05.860 --> 00:30:17.800 And each step, the energy goes up by some specific quantity. 00:30:22.920 --> 00:30:26.520 Therefore, because the energy cannot increase past 1, 00:30:26.520 --> 00:30:30.200 this iteration stops after a bounded number of steps. 00:30:33.160 --> 00:30:36.800 And once it's done, we end up with 00:30:36.800 --> 00:30:42.000 a epsilon regular partition. 00:30:42.000 --> 00:30:44.800 So that's the basic strategy. 00:30:44.800 --> 00:30:49.620 And what I want to show you is how to execute that strategy. 00:30:49.620 --> 00:30:50.986 Any questions so far? 00:30:50.986 --> 00:30:51.486 Yes. 00:30:51.486 --> 00:30:55.760 AUDIENCE: Just to clarify [INAUDIBLE] a bit, 00:30:55.760 --> 00:30:59.574 if some Vi's into non-epsilon regular partitions, 00:30:59.574 --> 00:31:04.347 is it possible for Ai,j and Aik to overlap somehow, right? 00:31:04.347 --> 00:31:07.502 Just kind of make those into three partitions? 00:31:07.502 --> 00:31:09.210 YUFEI ZHAO: So if I understand correctly, 00:31:09.210 --> 00:31:13.880 you are worried about between different pairs, 00:31:13.880 --> 00:31:15.110 you might have interactions. 00:31:15.110 --> 00:31:15.875 AUDIENCE: Yeah. 00:31:15.875 --> 00:31:17.500 YUFEI ZHAO: So you have seen the proof, 00:31:17.500 --> 00:31:19.917 but I think this is actually a very important and somewhat 00:31:19.917 --> 00:31:25.330 subtle point, is that I do not refine at each step, 00:31:25.330 --> 00:31:27.520 I find a pair of witnessing sets. 00:31:27.520 --> 00:31:31.860 I find all of these witnessing sets all at the same time, 00:31:31.860 --> 00:31:35.758 and I refine everything all at once. 00:31:35.758 --> 00:31:37.258 AUDIENCE: OK, so it's like if you do 00:31:37.258 --> 00:31:39.780 have overlap between two witnessing sets, that's OK? 00:31:39.780 --> 00:31:42.900 YUFEI ZHAO: That is OK, because this step doesn't care. 00:31:42.900 --> 00:31:45.720 If you have two witnessing sets that overlap, that is OK. 00:31:48.280 --> 00:31:49.210 We'll see the proof. 00:31:49.210 --> 00:31:49.815 Yes. 00:31:49.815 --> 00:31:51.273 AUDIENCE: Do you just find one pair 00:31:51.273 --> 00:31:53.652 of witnessing sets for each Vi, Vj, 00:31:53.652 --> 00:31:55.030 even though there might be more? 00:31:55.030 --> 00:31:56.488 YUFEI ZHAO: Question is, do we find 00:31:56.488 --> 00:31:58.235 just one pair of witnessing sets even 00:31:58.235 --> 00:31:59.360 though there could be more? 00:31:59.360 --> 00:32:00.100 And the answer is, yes. 00:32:00.100 --> 00:32:01.360 We just need to find one. 00:32:01.360 --> 00:32:02.430 There could be lots. 00:32:02.430 --> 00:32:04.600 So if it's not epsilon regular, it 00:32:04.600 --> 00:32:07.600 might be very not epsilon regular. 00:32:07.600 --> 00:32:11.450 And in fact, being a witnessing set is a fairly robust notion. 00:32:11.450 --> 00:32:13.700 If you just take out a small number of vertices, 00:32:13.700 --> 00:32:17.206 it's still a witnessing set. 00:32:17.206 --> 00:32:20.160 Any more questions? 00:32:20.160 --> 00:32:20.660 Great. 00:32:20.660 --> 00:32:25.090 So let's take a quick break and then we'll see the proof. 00:32:25.090 --> 00:32:28.260 Let's get started with the proof of Szemeredi's regularity 00:32:28.260 --> 00:32:29.050 lemma. 00:32:29.050 --> 00:32:31.420 And to do the proof, I want to develop 00:32:31.420 --> 00:32:34.510 this notion of energy which you saw in the proof sketch. 00:32:37.290 --> 00:32:39.830 So what do I mean by "energy?" 00:32:39.830 --> 00:32:46.640 First, if I-- let me define some quantities. 00:32:46.640 --> 00:32:53.870 If I have two vertex subsets, U and W, 00:32:53.870 --> 00:33:00.810 let me define this quantity, q, which is basically 00:33:00.810 --> 00:33:05.470 the edge density squared. 00:33:05.470 --> 00:33:08.580 But I normalize it somewhat according 00:33:08.580 --> 00:33:11.220 to how big U and W are. 00:33:14.560 --> 00:33:18.180 I'm going to use the letter and N to denote 00:33:18.180 --> 00:33:26.880 the number of vertices in G. So this is some cube. 00:33:26.880 --> 00:33:36.980 And for partitions, if I have a pair of partitions, 00:33:36.980 --> 00:33:56.230 Pu of U into k parts, and the partition Pw of W into l parts, 00:33:56.230 --> 00:34:14.770 I set this q of Pu and Pw to be the quantity where I sum over 00:34:14.770 --> 00:34:18.370 basically all pairs, one part from U, 00:34:18.370 --> 00:34:27.409 one part from W of this q between Ui and Wj. 00:34:30.150 --> 00:34:32.010 So this is the density squared. 00:34:32.010 --> 00:34:35.909 And I'm taking some kind of weighted average 00:34:35.909 --> 00:34:39.280 of the squared density. 00:34:39.280 --> 00:34:41.370 So here is a weighted average. 00:34:41.370 --> 00:34:44.280 If you prefer to think about the special case 00:34:44.280 --> 00:34:47.310 where this partition is an equipartition, 00:34:47.310 --> 00:34:51.750 then it is really the average of these squared densities. 00:34:51.750 --> 00:34:53.070 It's a mean square density. 00:34:55.860 --> 00:35:07.450 And finally, for a partition P of the vertex set of G 00:35:07.450 --> 00:35:18.560 into m parts, we define this q of this partition P 00:35:18.560 --> 00:35:23.480 to be q of P with itself according 00:35:23.480 --> 00:35:25.730 to the previous definition. 00:35:25.730 --> 00:35:32.000 Or in other words, I do this double sum, i from 1 to m, 00:35:32.000 --> 00:35:37.460 j from 1 to m, q of Vi, Vj. 00:35:40.550 --> 00:35:42.660 And this is the quantity that I will call 00:35:42.660 --> 00:35:53.300 the "energy" of the partition. 00:35:53.300 --> 00:36:01.440 It is a mean squared density, some weighted mean 00:36:01.440 --> 00:36:05.220 of the edge densities between pairs 00:36:05.220 --> 00:36:07.072 of parts in the partition. 00:36:09.970 --> 00:36:12.800 You might ask, why is it called an energy? 00:36:12.800 --> 00:36:15.370 So you might see from this formula here, 00:36:15.370 --> 00:36:19.250 it's some kind of a mean square density, 00:36:19.250 --> 00:36:22.600 so it's some kind of an average of squares. 00:36:22.600 --> 00:36:27.640 So in particular, it's some kind of an L2 quantity. 00:36:27.640 --> 00:36:31.900 And there's a general phenomenon in mathematics, 00:36:31.900 --> 00:36:35.110 I think borrowed from physical intuitions, 00:36:35.110 --> 00:36:37.390 that you can pretty much call anything that's 00:36:37.390 --> 00:36:40.900 an L2 quantity an energy. 00:36:40.900 --> 00:36:43.460 And so that's, I think, where the name comes from. 00:36:46.260 --> 00:36:48.920 So this is the important object for our proof. 00:36:48.920 --> 00:36:53.090 And let's see how to execute a strategy, the energy increment 00:36:53.090 --> 00:36:56.410 argument outlined on the board over there. 00:36:56.410 --> 00:37:01.060 So we want to show that you can refine a partition that 00:37:01.060 --> 00:37:06.100 is not epsilon regular in such a way that the energy goes up. 00:37:08.690 --> 00:37:11.960 And to do that, let me state a few lemmas regarding 00:37:11.960 --> 00:37:14.930 the energy of a partition under refinement. 00:37:24.230 --> 00:37:26.560 And the point of the next several lemmas 00:37:26.560 --> 00:37:42.170 is that the energy never decreases under refinement, 00:37:42.170 --> 00:37:48.020 and it sometimes increases if your partition is not 00:37:48.020 --> 00:37:48.860 epsilon regular. 00:37:51.460 --> 00:37:54.380 So the first lemma is that if you 00:37:54.380 --> 00:38:03.340 look at the energy between a pair of partitions, 00:38:03.340 --> 00:38:11.730 it is never less than the energy between the two vertex sets. 00:38:11.730 --> 00:38:22.540 So for instance, if you have U and W like that, 00:38:22.540 --> 00:38:28.380 and I partition them into Pu and Pw, 00:38:28.380 --> 00:38:31.000 and I measure the energy, just basically 00:38:31.000 --> 00:38:35.620 the squared density between U and V versus summing up 00:38:35.620 --> 00:38:38.700 the individual squared densities after the partition, 00:38:38.700 --> 00:38:41.200 the left side is always at least as great as the right side. 00:38:45.110 --> 00:38:47.080 So this is really a claim. 00:38:47.080 --> 00:38:49.925 It's a fairly simple claim about convexity, 00:38:49.925 --> 00:38:51.550 but let me set it up in a way that will 00:38:51.550 --> 00:38:54.440 help some of the later proofs. 00:38:54.440 --> 00:39:00.900 So let me define a random variable, which 00:39:00.900 --> 00:39:02.772 I call Z, in the following way. 00:39:02.772 --> 00:39:04.230 So here's a process that I will use 00:39:04.230 --> 00:39:06.510 to define this random variable. 00:39:06.510 --> 00:39:12.540 I will select x, little x, to be a vertex uniformly chosen 00:39:12.540 --> 00:39:16.160 from U, from the left vertex set. 00:39:16.160 --> 00:39:27.670 And I will select a vertex y uniformly chosen from W. 00:39:27.670 --> 00:39:34.540 x and y, they fall into some part in the partition. 00:39:34.540 --> 00:39:42.630 So suppose Ui is the part where x i 00:39:42.630 --> 00:39:52.450 falls, and Wi is the set in the partition where y falls. 00:39:52.450 --> 00:39:55.840 So Ui is a member of this partition. 00:39:55.840 --> 00:40:00.500 Wi is a member of the other partition of W. 00:40:00.500 --> 00:40:03.980 Then I define my random variable Z 00:40:03.980 --> 00:40:11.870 to be the x density between Ui and Wj. 00:40:11.870 --> 00:40:12.500 So it's Wj. 00:40:17.030 --> 00:40:19.170 So that's the definition. 00:40:19.170 --> 00:40:21.500 So pick x randomly. 00:40:21.500 --> 00:40:22.850 Pick y randomly. 00:40:22.850 --> 00:40:25.010 Suppose x falls in Ui. 00:40:25.010 --> 00:40:27.140 Suppose y falls in Uj. 00:40:27.140 --> 00:40:31.610 Then Z is the x density between these two parts. 00:40:34.170 --> 00:40:37.120 So Z is some random variable. 00:40:37.120 --> 00:40:41.700 Let's look at properties of this random variable. 00:40:41.700 --> 00:40:45.947 First, what is this, it's expectation? 00:40:45.947 --> 00:40:47.280 It's a discrete random variable. 00:40:47.280 --> 00:40:49.363 And you can easily compute all of these quantities 00:40:49.363 --> 00:40:54.670 by just summing up according to how Z is generated. 00:40:54.670 --> 00:40:57.540 So I look overall, i and j. 00:40:57.540 --> 00:41:01.430 What's the probability that x falls in Ui? 00:41:01.430 --> 00:41:04.680 It is the size of Ui as a fraction of U. 00:41:04.680 --> 00:41:07.110 What's the probability that y falls in Wj? 00:41:07.110 --> 00:41:13.080 It's the size of Wj as a fraction of size W. 00:41:13.080 --> 00:41:17.460 And then Z is this quantity here. 00:41:17.460 --> 00:41:24.390 So this is what I find to be the expectation of Z. 00:41:24.390 --> 00:41:27.580 But you see the density multiplied by the product 00:41:27.580 --> 00:41:31.090 of the vertex set sizes, that's just the number of edges 00:41:31.090 --> 00:41:38.980 between U and W. And you sum over all the i, j's. 00:41:38.980 --> 00:41:50.080 So that, which is simply the edge density between U and W. 00:41:50.080 --> 00:41:53.570 So that's the expectation of the Z variable. 00:41:53.570 --> 00:41:56.520 On the other hand, what's the second moment? 00:41:56.520 --> 00:42:01.040 In other words, what's the expectation of the square of Z? 00:42:01.040 --> 00:42:05.080 Again, we do the same computation. 00:42:05.080 --> 00:42:06.280 First part is the same. 00:42:11.610 --> 00:42:13.760 The second part now becomes a d squared. 00:42:19.610 --> 00:42:27.070 And look at how we define energy. 00:42:27.070 --> 00:42:32.960 This quantity here is basically the energy q 00:42:32.960 --> 00:42:37.460 between the partition U and the partition of W, 00:42:37.460 --> 00:42:40.630 except there's normalization. 00:42:40.630 --> 00:42:43.140 That's not quite the same as the one we used before. 00:42:43.140 --> 00:42:45.620 So we will just put in that normalization. 00:42:54.740 --> 00:42:57.440 So now you compare the expectation 00:42:57.440 --> 00:43:01.540 of Z versus the expectation of Z squared. 00:43:01.540 --> 00:43:09.030 And we know by convexity that the expectation of Z squared 00:43:09.030 --> 00:43:13.530 is at least as large as the expectation of Z, 00:43:13.530 --> 00:43:14.640 that quantity squared. 00:43:17.400 --> 00:43:22.710 But if you plug in what values you get for these two guys, 00:43:22.710 --> 00:43:29.140 you derive the inequality claimed in Lemma 1. 00:43:32.260 --> 00:43:34.300 You have to cancel some normalization factors, 00:43:34.300 --> 00:43:35.217 but that's easy to do. 00:43:37.717 --> 00:43:38.800 So that's the first lemma. 00:43:43.060 --> 00:43:46.230 So the first one is just about a pair of parts, 00:43:46.230 --> 00:43:49.330 and identify partition, each part, 00:43:49.330 --> 00:43:52.030 what happens to the energy between this pair. 00:43:52.030 --> 00:43:54.820 And the second one is a direct corollary of the first one. 00:43:54.820 --> 00:43:57.760 It says that if you have a second partition, 00:43:57.760 --> 00:44:08.660 P prime that refines P, then the energy of the second partition, 00:44:08.660 --> 00:44:11.510 the refinement, is never less than the energy 00:44:11.510 --> 00:44:12.887 of the first partition. 00:44:15.570 --> 00:44:18.660 And it is a direct consequence of the first lemma, 00:44:18.660 --> 00:44:32.790 because we simply apply this lemma to every pair of parts 00:44:32.790 --> 00:44:39.890 in P. Between every pair of parts, 00:44:39.890 --> 00:44:41.520 the energy can never go down. 00:44:41.520 --> 00:44:43.600 So overall, the energy does not go down. 00:44:48.150 --> 00:44:52.170 And finally, so far, we've just said that the partitions can 00:44:52.170 --> 00:44:54.550 never make the energy go down. 00:44:54.550 --> 00:44:56.190 But in order to do this proof, we 00:44:56.190 --> 00:44:58.740 need to show that the energy sometimes goes up. 00:45:01.203 --> 00:45:02.870 And that's the point of the third lemma. 00:45:02.870 --> 00:45:10.400 The third Lemma tells us that you can get an energy boost. 00:45:10.400 --> 00:45:12.020 So this is the Red Bull Lemma. 00:45:12.020 --> 00:45:16.962 You can get an energy boost if you are feeling irregular. 00:45:20.820 --> 00:45:28.360 So if U, W is not epsilon regular, 00:45:28.360 --> 00:45:37.270 and this epsilon regularity is witnessed by U1 in U and W1 00:45:37.270 --> 00:45:51.070 in W, then I claim that the energy obtained by chopping U 00:45:51.070 --> 00:46:00.260 into U1 and its complement against W1, 00:46:00.260 --> 00:46:10.260 against the complement of W1 and W, so here again U and W, 00:46:10.260 --> 00:46:15.830 I find a witnessing set for their irregularity. 00:46:15.830 --> 00:46:18.980 And now I partition left and right according to-- 00:46:18.980 --> 00:46:22.190 chop each part into two. 00:46:22.190 --> 00:46:25.900 So this energy between this partition 00:46:25.900 --> 00:46:33.670 into two on both sides is bigger than the original energy 00:46:33.670 --> 00:46:38.280 plus something where we can gain. 00:46:38.280 --> 00:46:41.660 And this something where we can gain turns out 00:46:41.660 --> 00:46:46.050 to be at least epsilon raised to the power 00:46:46.050 --> 00:46:52.095 4 times the size of U, size of W, divided by n squared. 00:47:02.860 --> 00:47:03.700 Can you prove it? 00:47:08.190 --> 00:47:15.250 Let's define Z the same as in the previous proof, 00:47:15.250 --> 00:47:17.550 as in the proof of Lemma 1. 00:47:22.180 --> 00:47:26.070 In Lemma 1, we just used the fact that the L2 norm of Z, 00:47:26.070 --> 00:47:28.985 the expectation of the square, is at least 00:47:28.985 --> 00:47:30.610 the square of the expectation. 00:47:30.610 --> 00:47:32.360 But actually, there are differences in it. 00:47:32.360 --> 00:47:35.108 It's called a "variance." 00:47:35.108 --> 00:47:38.780 The variance of Z is the difference 00:47:38.780 --> 00:47:41.340 between these two quantities. 00:47:41.340 --> 00:47:44.852 I know that it's always non-negative. 00:47:47.810 --> 00:47:51.115 So if you look at how we derived the expectation of Z 00:47:51.115 --> 00:47:53.320 and expectation of Z squared, you immediately 00:47:53.320 --> 00:48:01.080 see that its variance we can write 00:48:01.080 --> 00:48:07.230 as, up to a normalizing factor, the difference 00:48:07.230 --> 00:48:20.090 between this energy on one hand and the energy between U and W, 00:48:20.090 --> 00:48:26.380 namely the mean square of the normalization. 00:48:26.380 --> 00:48:31.810 On the other hand, a different way to calculate the variance 00:48:31.810 --> 00:48:38.500 is that it is equal to the expectation of the deviation 00:48:38.500 --> 00:48:40.150 from the mean squared. 00:48:44.310 --> 00:48:47.320 So let's think about the deviation from the mean. 00:48:47.320 --> 00:48:53.940 I am choosing a random vector in, on the left, U, 00:48:53.940 --> 00:48:57.240 and another random point, random vertex on the left, 00:48:57.240 --> 00:49:00.570 and a random point on the right. 00:49:00.570 --> 00:49:05.220 In the event that they both lie in the sets 00:49:05.220 --> 00:49:17.110 that witness the irregularity, so in the event 00:49:17.110 --> 00:49:21.430 where x falls here and y falls here, 00:49:21.430 --> 00:49:24.930 which occurs with this probability, 00:49:24.930 --> 00:49:36.310 see that this quantity here is equal to the density between U1 00:49:36.310 --> 00:49:40.730 and W1 minus the density-- 00:49:40.730 --> 00:49:44.470 and this expectation of Z is just 00:49:44.470 --> 00:49:50.710 the density between U and W. So interpreting this expectation 00:49:50.710 --> 00:49:55.750 for what happens when x falls in U1 00:49:55.750 --> 00:50:02.210 and when y falls in W1, ignoring all the other events, 00:50:02.210 --> 00:50:06.230 because the quantity is always non-negative everywhere. 00:50:06.230 --> 00:50:14.660 But now from the definition of epsilon regularity, 00:50:14.660 --> 00:50:19.010 or rather the witnessing of epsilon irregularity, 00:50:19.010 --> 00:50:26.560 you see that this U1 is at least an epsilon fraction of U. W1's 00:50:26.560 --> 00:50:29.800 at least an epsilon fraction of W. 00:50:29.800 --> 00:50:35.500 And this final quantity here is at least epsilon inside, so 00:50:35.500 --> 00:50:38.600 at least epsilon squared. 00:50:38.600 --> 00:50:41.110 So here we're using all the different components 00:50:41.110 --> 00:50:43.270 of the definition of epsilon regular. 00:50:43.270 --> 00:50:44.506 Yes. 00:50:44.506 --> 00:50:48.988 AUDIENCE: What happens if we're dividing with more witnessing 00:50:48.988 --> 00:50:50.705 sets? 00:50:50.705 --> 00:50:52.080 YUFEI ZHAO: So you're asking what 00:50:52.080 --> 00:50:54.150 happens if we divide with more witnessing sets? 00:50:54.150 --> 00:50:55.650 So hold onto that thought. 00:50:55.650 --> 00:50:58.050 So right now, I'm just showing what happens 00:50:58.050 --> 00:50:59.640 if you have one witnessing set. 00:51:02.250 --> 00:51:04.590 Any more questions? 00:51:04.590 --> 00:51:08.070 So here we have epsilon to the 4th. 00:51:08.070 --> 00:51:11.010 And if you're putting the normalization 00:51:11.010 --> 00:51:14.180 comparing these two interpretations, 00:51:14.180 --> 00:51:16.490 you'll find the inequality claimed by the lemma. 00:51:23.180 --> 00:51:28.470 So now we are ready to show the key part of this iteration. 00:51:31.080 --> 00:51:33.720 I'll show you precisely how this iteration works, and show 00:51:33.720 --> 00:51:39.270 that you always get an energy boost in the overall partition. 00:51:39.270 --> 00:51:43.110 So I'll call the next one Lemma 4. 00:51:43.110 --> 00:51:52.350 And this says that if you have a partition P of the vertex set 00:51:52.350 --> 00:52:01.040 of G into k parts, if this partition is not epsilon 00:52:01.040 --> 00:52:21.900 regular, then there exists a refinement called Q where every 00:52:21.900 --> 00:52:35.720 part V sub i is partitioned further into at most 2 the k 00:52:35.720 --> 00:52:46.180 parts, and such that the partition of Q-- 00:52:46.180 --> 00:52:49.070 so the energy of the new partition Q-- 00:52:49.070 --> 00:52:54.800 increases substantially from the previous partition P. 00:52:54.800 --> 00:52:56.300 And we'll show that you can increase 00:52:56.300 --> 00:53:02.690 by at least epsilon to the 5th power, 00:53:02.690 --> 00:53:05.133 some constant in epsilon. 00:53:05.133 --> 00:53:06.800 So if you look at the strategy up there, 00:53:06.800 --> 00:53:08.960 if you can do this every step, then that 00:53:08.960 --> 00:53:10.610 means that the number of iterations 00:53:10.610 --> 00:53:13.670 is bounded by 1 over epsilon to the 5th power. 00:53:22.580 --> 00:53:27.480 So to prove this lemma here, we will use the three lemmas 00:53:27.480 --> 00:53:29.580 up there and put them together. 00:53:35.900 --> 00:53:43.710 So for all the pairs i, j such that V sub i, 00:53:43.710 --> 00:53:51.310 V sub j is not epsilon regular, as outlined 00:53:51.310 --> 00:53:53.800 in the proof in the outline up there, 00:53:53.800 --> 00:54:02.350 we will find this A superscript i, j in Vi, 00:54:02.350 --> 00:54:11.869 and A superscript j, i in Vj that witness the irregularity. 00:54:17.250 --> 00:54:23.180 So do this simultaneously for all pairs i, comma, j, 00:54:23.180 --> 00:54:25.550 where the Vi, Vj is not epsilon regular. 00:54:28.390 --> 00:54:34.900 Now what we're going to define Q as is the common refinement. 00:54:34.900 --> 00:54:37.390 So take all of, just as indicated 00:54:37.390 --> 00:54:39.550 in that picture up there, simultaneously 00:54:39.550 --> 00:54:55.040 take all of these A's and use them to refine P. Starting 00:54:55.040 --> 00:54:57.080 with P, starting with a partition you have, 00:54:57.080 --> 00:54:59.750 simultaneously cut everything up using 00:54:59.750 --> 00:55:01.240 all of these witnessing sets. 00:55:05.980 --> 00:55:09.560 Now, we only have witnessing pairs for pairs 00:55:09.560 --> 00:55:10.970 that are not epsilon regular. 00:55:10.970 --> 00:55:13.262 If they're epsilon regular, you don't worry about them. 00:55:19.570 --> 00:55:21.390 One of the claims in the lemma now is-- 00:55:21.390 --> 00:55:23.160 this is the Q that we'll end up with. 00:55:23.160 --> 00:55:25.470 We'll show that this Q has that property. 00:55:25.470 --> 00:55:26.880 So one of the claims in the lemma 00:55:26.880 --> 00:55:33.390 is that every Vi is partitioned into at most 2 to the k parts. 00:55:33.390 --> 00:55:37.710 So I hope that part is clear, because how are we 00:55:37.710 --> 00:55:39.360 doing the refinement? 00:55:39.360 --> 00:55:40.520 We're taking Vi. 00:55:44.810 --> 00:55:57.610 It's divided into parts using these A i, 00:55:57.610 --> 00:56:06.850 j's, one j coming from each pair that is irregular with Vi. 00:56:06.850 --> 00:56:12.500 So I'm cutting up Vi using at most k sets, 00:56:12.500 --> 00:56:15.090 so one coming from each of the other possible. 00:56:15.090 --> 00:56:18.090 Maybe fewer than k-- that's fine-- but at most k sets 00:56:18.090 --> 00:56:20.630 are used to cut up each Vi. 00:56:20.630 --> 00:56:24.355 So you have at most 2 to the k parts 00:56:24.355 --> 00:56:25.480 once you cut everything up. 00:56:30.820 --> 00:56:39.570 But the tricky part is to show that you get an energy boost. 00:56:39.570 --> 00:56:40.320 So let's do this. 00:56:40.320 --> 00:56:42.153 How do we show that you get an energy boost? 00:56:49.720 --> 00:56:52.020 We're going to put the top three lemmas together. 00:56:54.864 --> 00:57:01.510 First, we want to analyze the energy of Q. 00:57:01.510 --> 00:57:03.320 So let's write it out. 00:57:03.320 --> 00:57:12.790 So the energy of Q is the sum over this energy of individual 00:57:12.790 --> 00:57:14.020 partitions of the Vi's. 00:57:17.810 --> 00:57:28.280 And by this P sub Vi, P sub Vj, I mean the partition of Vj 00:57:28.280 --> 00:57:36.060 given by Q. So what happens after you cut up the Vi-- 00:57:36.060 --> 00:57:43.170 that's what I mean by P sub Vj, or rather I should call it 00:57:43.170 --> 00:57:45.070 Q sub Vi, Q sub Vj. 00:57:52.260 --> 00:58:01.830 By Lemma 2, we find that-- 00:58:01.830 --> 00:58:04.490 so let me separate them into two cases. 00:58:04.490 --> 00:58:10.650 The first case sums over i, j such that Vi, 00:58:10.650 --> 00:58:16.140 Vj is epsilon regular. 00:58:16.140 --> 00:58:22.620 And by Lemma 1, so here we're using Lemma 1, 00:58:22.620 --> 00:58:29.130 we find that this quantity here cannot be less than the Q 00:58:29.130 --> 00:58:31.580 of Vi, Vj. 00:58:31.580 --> 00:58:35.510 So take those two parts. 00:58:35.510 --> 00:58:38.680 Before and after the refinement by Q, 00:58:38.680 --> 00:58:39.830 the energy cannot go down. 00:58:43.660 --> 00:58:47.780 So I don't worry too much about pairs that are epsilon regular. 00:58:47.780 --> 00:58:51.900 But no let me look up here that are not epsilon regular. 00:59:05.630 --> 00:59:09.685 So what we will do now is even though-- 00:59:09.685 --> 00:59:11.310 so let's look at that picture up there. 00:59:15.750 --> 00:59:20.970 So let's focus on what I drew in red. 00:59:20.970 --> 00:59:24.670 So let's focus between 1 and 2. 00:59:24.670 --> 00:59:32.280 So suppose the shaded part is the witnessing sets. 00:59:32.280 --> 00:59:34.890 The witnessing sets got cut up further 00:59:34.890 --> 00:59:39.490 by other witnessing sets. 00:59:39.490 --> 00:59:46.810 But I don't have to worry about them because Lemma 2 or Lemma 00:59:46.810 --> 00:59:53.110 1, really, tells me that I can do an inequality where 00:59:53.110 --> 01:00:00.940 I go down to just comparing the energy between this partition 01:00:00.940 --> 01:00:05.030 of two parts, this single witnessing set 01:00:05.030 --> 01:00:16.250 and its complement, versus what happens in its partner. 01:00:24.620 --> 01:00:32.600 So in other words, over here, the Q of this pair, 01:00:32.600 --> 01:00:40.670 I am saying that it is no less than if I just 01:00:40.670 --> 01:00:47.280 look at what happens if you only cut up these two sets using 01:00:47.280 --> 01:00:47.950 the red lines. 01:00:55.790 --> 01:00:57.960 Let's go on. 01:00:57.960 --> 01:01:02.400 Applying Lemma 3, the energy boost lemma, 01:01:02.400 --> 01:01:04.960 the first part stays the same. 01:01:04.960 --> 01:01:06.820 So this first part stays the same. 01:01:06.820 --> 01:01:08.640 And the second part, now, because I'm 01:01:08.640 --> 01:01:11.160 looking at witnessing sets for irregularity, 01:01:11.160 --> 01:01:12.630 I get this extra boost. 01:01:15.950 --> 01:01:19.240 So this goes back to one of the questions asked earlier, 01:01:19.240 --> 01:01:23.260 where in Lemma 3, I don't have to now worry about what happens 01:01:23.260 --> 01:01:28.780 if you have further cuts, because I only need to worry 01:01:28.780 --> 01:01:31.240 about the case where I only have a single cut 01:01:31.240 --> 01:01:34.740 between the epsilon irregular pairs. 01:01:34.740 --> 01:01:39.900 So putting it together, we see that the previous line 01:01:39.900 --> 01:01:48.740 is at least, if you sum over the Q's of all the pairs 01:01:48.740 --> 01:01:59.420 plus this extra epsilon to the 4th term for all pairs 01:01:59.420 --> 01:02:01.540 that are not epsilon regular. 01:02:17.890 --> 01:02:23.840 I'm applying monotonicity of energy for the types, 01:02:23.840 --> 01:02:25.760 for pairs that are epsilon regular, 01:02:25.760 --> 01:02:29.080 an energy boost for pairs that are not epsilon regular. 01:02:29.080 --> 01:02:33.118 And for the latter type, I obtain this boost. 01:02:33.118 --> 01:02:35.160 Now remember what's the definition of an "epsilon 01:02:35.160 --> 01:02:36.270 regular partition." 01:02:36.270 --> 01:02:38.100 Unfortunately, it's no longer on the board, 01:02:38.100 --> 01:02:42.890 but it says that this sum over here, 01:02:42.890 --> 01:02:44.990 if it is an epsilon regular partition, 01:02:44.990 --> 01:02:47.660 it is at most epsilon. 01:02:47.660 --> 01:02:51.980 So if it is not epsilon regular, we can lower bound it. 01:02:51.980 --> 01:02:55.680 And that's indeed what we will do. 01:02:55.680 --> 01:02:58.050 The first sum here is, by definition, 01:02:58.050 --> 01:03:04.020 Q of the partition P. And the second sum, 01:03:04.020 --> 01:03:07.890 by the definition of epsilon regular, 01:03:07.890 --> 01:03:12.840 is at least epsilon to the power 5. 01:03:12.840 --> 01:03:14.480 So here we're using the definition 01:03:14.480 --> 01:03:17.590 of epsilon regular partition, namely, 01:03:17.590 --> 01:03:22.330 that a large fraction, so at least an epsilon 01:03:22.330 --> 01:03:28.240 fraction, basically, of pairs of vertex sets 01:03:28.240 --> 01:03:34.940 are not epsilon regular, but in this weighted sense. 01:03:34.940 --> 01:03:38.106 And that finishes the proof of Lemma 4 up there. 01:03:41.760 --> 01:03:42.930 Any questions so far? 01:03:47.650 --> 01:03:50.890 All right, so now we are ready to finish everything 01:03:50.890 --> 01:03:53.750 off, and prove Szemeredi's regularity lemma. 01:04:21.530 --> 01:04:28.280 So let's prove Szemeredi's regularity lemma. 01:04:33.480 --> 01:04:43.160 Let's start with the trivial partition, 01:04:43.160 --> 01:04:46.250 meaning just one large part. 01:04:46.250 --> 01:04:53.880 And we are going to repeatedly apply 01:04:53.880 --> 01:05:01.060 Lemma 4 whenever the partition at hand 01:05:01.060 --> 01:05:14.290 is not regular, whenever the current partition is not 01:05:14.290 --> 01:05:15.190 epsilon regular. 01:05:20.600 --> 01:05:22.930 So let's look at its energy. 01:05:22.930 --> 01:05:28.590 The energy of this partition-- 01:05:28.590 --> 01:05:34.860 so this is a weighted mean of the edge density squared, 01:05:34.860 --> 01:05:40.020 so it always lies between 0 and 1, 01:05:40.020 --> 01:05:44.760 just from the definition of energy. 01:05:44.760 --> 01:05:51.990 On the other hand, Lemma 4 tells us 01:05:51.990 --> 01:06:00.840 that the energy increases by at least epsilon to the 5th power 01:06:00.840 --> 01:06:03.304 at each iteration. 01:06:10.570 --> 01:06:14.370 So this process cannot continue forever. 01:06:14.370 --> 01:06:23.408 So it must stop after at most epsilon to the minus 5th power 01:06:23.408 --> 01:06:24.075 number of steps. 01:06:27.080 --> 01:06:32.590 And when we stop, we must result in an epsilon regular 01:06:32.590 --> 01:06:39.220 partition, because otherwise, you're 01:06:39.220 --> 01:06:43.156 going to continue applying the lemma and push it even further. 01:06:43.156 --> 01:06:46.240 And that's it. 01:06:46.240 --> 01:06:50.500 So that proves Szemeredi's graph regularity lemma. 01:06:53.440 --> 01:06:54.910 Question. 01:06:54.910 --> 01:06:57.360 AUDIENCE: It's going to be some really big value of M. 01:06:57.360 --> 01:06:59.030 YUFEI ZHAO: OK, let's talk about bounds. 01:06:59.030 --> 01:07:00.490 So let's talk about how many parts. 01:07:06.580 --> 01:07:11.410 So how many parts does this proof produce? 01:07:11.410 --> 01:07:12.350 We can figure it out. 01:07:12.350 --> 01:07:14.470 So we have some number of steps. 01:07:14.470 --> 01:07:18.440 Each step increases the number of parts by something. 01:07:18.440 --> 01:07:30.840 So if P has k parts, so then Lemma 4 refines 01:07:30.840 --> 01:07:37.910 P into at most how many parts? 01:07:37.910 --> 01:07:38.823 AUDIENCE: 2 to the k 01:07:38.823 --> 01:07:40.490 YUFEI ZHAO: Yeah, so k times 2 to the k. 01:07:44.330 --> 01:07:48.720 And I have many iterations of this guy. 01:07:48.720 --> 01:07:50.220 So some of you are already laughing, 01:07:50.220 --> 01:07:53.280 because it's going to be a very large number. 01:07:53.280 --> 01:07:55.310 In fact, because it's going to be so large, 01:07:55.310 --> 01:07:57.460 it makes my calculations slightly more convenient. 01:07:57.460 --> 01:07:58.918 It really doesn't change the answer 01:07:58.918 --> 01:08:01.820 so much if I just bound k to the 2 to the k by 2 to the 2 01:08:01.820 --> 01:08:03.050 to the k. 01:08:03.050 --> 01:08:10.440 So the final number of parts is this function 01:08:10.440 --> 01:08:15.030 iterated on itself epsilon to the minus 5 times. 01:08:15.030 --> 01:08:22.590 So it's a power of 2 of height at most 2 to the epsilon 01:08:22.590 --> 01:08:23.090 to the 5. 01:08:26.560 --> 01:08:29.477 It's a finite number, so it depends only on epsilon and not 01:08:29.477 --> 01:08:30.560 on the size of your graph. 01:08:30.560 --> 01:08:32.102 And this is the most important thing. 01:08:32.102 --> 01:08:35.290 It does not depend on the size of your graph. 01:08:35.290 --> 01:08:38.020 It is quite large. 01:08:38.020 --> 01:08:42.880 In fact, even for reasonable values of epsilon, 01:08:42.880 --> 01:08:48.850 like 1% or even 10%, this number is astronomically large. 01:08:48.850 --> 01:08:51.490 And you may ask is it really necessary, 01:08:51.490 --> 01:08:54.372 because we did this proof, and it came out fairly elegantly, 01:08:54.372 --> 01:08:56.080 I would say it, how the proof was set up. 01:08:56.080 --> 01:08:58.779 And you arrived at this finite bound. 01:08:58.779 --> 01:09:00.319 But maybe there's a better proof. 01:09:00.319 --> 01:09:04.170 Maybe you can work harder and obtain somewhat better bounds. 01:09:04.170 --> 01:09:08.250 So you can ask, is it possible that the truth is really 01:09:08.250 --> 01:09:11.103 somehow much smaller? 01:09:11.103 --> 01:09:12.520 And the answer turns out to be no. 01:09:15.189 --> 01:09:21.100 So there is a theorem by Tim Gowers 01:09:21.100 --> 01:09:24.087 which says that there exists some constant. 01:09:24.087 --> 01:09:26.170 The precise statement, again, is not so important, 01:09:26.170 --> 01:09:28.899 but based on what I just said, you cannot improve this bound 01:09:28.899 --> 01:09:31.180 given by this proof. 01:09:31.180 --> 01:09:38.910 So for every epsilon small enough, 01:09:38.910 --> 01:09:48.470 there exists a graph whose epsilon regular partition 01:09:48.470 --> 01:09:54.120 requires how many parts? 01:09:54.120 --> 01:10:02.950 So the number of parts at least this tower of 2 of height 01:10:02.950 --> 01:10:06.010 some epsilon to the minus c. 01:10:09.630 --> 01:10:15.200 So really it's a tower of exponentials of size, 01:10:15.200 --> 01:10:18.800 essentially polynomial in 1 over epsilon. 01:10:18.800 --> 01:10:22.250 So maybe you can squeeze the 5 to something less. 01:10:22.250 --> 01:10:24.250 Actually, we don't even know if that's the case, 01:10:24.250 --> 01:10:28.700 but certainly you cannot do substantially better than what 01:10:28.700 --> 01:10:31.530 the proof gives. 01:10:31.530 --> 01:10:36.600 So Szemeredi's regularity lemma is an extremely powerful tool. 01:10:36.600 --> 01:10:41.550 And we'll see applications that are basically 01:10:41.550 --> 01:10:42.560 very difficult to prove. 01:10:42.560 --> 01:10:44.260 And for some of these applications, 01:10:44.260 --> 01:10:45.960 we don't really know other proofs 01:10:45.960 --> 01:10:48.760 except using Szemeredi's regularity lemma. 01:10:48.760 --> 01:10:52.680 But on the other hand, it gives terrible quantitative bounds. 01:10:52.680 --> 01:10:57.450 So there is a lot of interest in combinatorics 01:10:57.450 --> 01:11:01.400 where once you see a proof that requires Szemeredi's regularity 01:11:01.400 --> 01:11:04.020 lemma, or that is first proved using this technique, 01:11:04.020 --> 01:11:07.870 to ask can it be used using some other technique? 01:11:07.870 --> 01:11:12.270 In fact, Szemeredi himself has worked a lot in that direction, 01:11:12.270 --> 01:11:14.854 trying to get rid of the uses of his lemma. 01:11:19.500 --> 01:11:22.410 Any questions? 01:11:22.410 --> 01:11:25.120 AUDIENCE: How could you modify it for equipartitions? 01:11:25.120 --> 01:11:26.170 YUFEI ZHAO: OK, great. 01:11:26.170 --> 01:11:29.038 Question is, how can we modify it for equipartitions? 01:11:29.038 --> 01:11:30.080 So let's talk about that. 01:11:30.080 --> 01:11:32.370 So it's a fantastic question. 01:11:32.370 --> 01:11:34.420 So look at this proof and see what can we 01:11:34.420 --> 01:11:37.720 do if we really want all the parts to have 01:11:37.720 --> 01:11:41.280 roughly the same size, let's say differing by at most 1. 01:12:00.800 --> 01:12:09.929 So how to make the epsilon regular partition equitable? 01:12:14.920 --> 01:12:16.220 Any guesses? 01:12:16.220 --> 01:12:18.810 Any attempts on what we can do? 01:12:18.810 --> 01:12:21.920 I mean, basically it's going to follow this proof. 01:12:21.920 --> 01:12:24.410 As I said, the spirit of Szemeredi's regularity lemma 01:12:24.410 --> 01:12:25.630 is what I've shown you. 01:12:25.630 --> 01:12:28.640 But the details and executions may vary somewhat 01:12:28.640 --> 01:12:31.210 depending on the specific purpose you have in mind. 01:12:31.210 --> 01:12:31.792 Yeah. 01:12:31.792 --> 01:12:33.238 AUDIENCE: Can we just add-- 01:12:33.238 --> 01:12:35.166 [INAUDIBLE] add things to the smaller part 01:12:35.166 --> 01:12:38.058 because we know that-- by the fact 01:12:38.058 --> 01:12:41.440 that it's not [INAUDIBLE] that parts aren't too small? 01:12:41.440 --> 01:12:44.510 YUFEI ZHAO: OK, so you're saying we're going to add something 01:12:44.510 --> 01:12:46.790 or to massage the partition to make it epsilon-- 01:12:46.790 --> 01:12:48.870 AUDIENCE: Add vertices to the smaller parts of the partition. 01:12:48.870 --> 01:12:51.495 YUFEI ZHAO: Add vertices to the smaller parts of the partition, 01:12:51.495 --> 01:12:53.828 now when are you going to do that? 01:12:53.828 --> 01:12:56.323 AUDIENCE: When they're-- like so you do the refinement, 01:12:56.323 --> 01:12:58.183 then when they're not [INAUDIBLE] 01:12:58.183 --> 01:12:59.600 YUFEI ZHAO: So you want to do this 01:12:59.600 --> 01:13:01.337 at every stage of the process. 01:13:01.337 --> 01:13:01.920 AUDIENCE: Yes. 01:13:01.920 --> 01:13:04.500 [INAUDIBLE] 01:13:04.500 --> 01:13:06.100 YUFEI ZHAO: I like that idea. 01:13:06.100 --> 01:13:09.920 So here's what we're going to do. 01:13:09.920 --> 01:13:12.410 So we still run the same process. 01:13:12.410 --> 01:13:15.760 So we're going to have this P, which is the current partition. 01:13:15.760 --> 01:13:20.980 So I have current partition. 01:13:20.980 --> 01:13:24.730 And as before, we initially have it 01:13:24.730 --> 01:13:27.420 as either the trivial partition, if you like, 01:13:27.420 --> 01:13:32.160 or m arbitrary equitable parts. 01:13:37.418 --> 01:13:39.210 Start with something where you don't really 01:13:39.210 --> 01:13:41.940 care about anything except for the size. 01:13:41.940 --> 01:13:46.080 And you run basically the same proof, 01:13:46.080 --> 01:13:53.000 where if your P is not epsilon regular, 01:13:53.000 --> 01:13:56.750 then do what we've done before, so basically 01:13:56.750 --> 01:13:58.160 exactly the same thing. 01:13:58.160 --> 01:14:13.590 We refine P using pairs witnessing regularity, 01:14:13.590 --> 01:14:15.750 same as the proof that we just did. 01:14:15.750 --> 01:14:17.910 And now we need to do something a little bit more 01:14:17.910 --> 01:14:21.480 to obtain equitability. 01:14:21.480 --> 01:14:24.660 And what we will do is right after-- 01:14:24.660 --> 01:14:29.180 so each step in iteration, right after we do this refinement, 01:14:29.180 --> 01:14:36.180 so after we cut up our graph where maybe some of the parts 01:14:36.180 --> 01:14:45.560 are really tiny, let's massage the partitions somewhat 01:14:45.560 --> 01:14:47.210 to make them equitable. 01:14:47.210 --> 01:14:51.000 And to make our life a little bit easier, 01:14:51.000 --> 01:14:56.130 we can refine the partition somewhat further 01:14:56.130 --> 01:15:00.720 to chop it up into somewhat smaller resolution. 01:15:00.720 --> 01:15:03.050 And this part, you can really do it 01:15:03.050 --> 01:15:04.370 either arbitrarily or randomly. 01:15:06.950 --> 01:15:10.888 Some ways may be slightly easier to execute, 01:15:10.888 --> 01:15:12.680 but it doesn't really matter how you do it. 01:15:12.680 --> 01:15:13.970 It's fairly robust. 01:15:13.970 --> 01:15:16.460 You refine it further. 01:15:16.460 --> 01:15:20.500 And basically, I want to make it equitable. 01:15:20.500 --> 01:15:23.260 Sometimes, you can just do that by refining, 01:15:23.260 --> 01:15:25.810 but maybe if you have some really small parts, 01:15:25.810 --> 01:15:28.300 then you might need to move some vertices around, 01:15:28.300 --> 01:15:31.150 so I call that "rebalancing." 01:15:34.040 --> 01:15:41.600 So move and merge some vertices, but only a very small number 01:15:41.600 --> 01:15:45.578 of vertices, to make equitable. 01:15:50.960 --> 01:15:55.713 So you run this loop until you find that your partition is 01:15:55.713 --> 01:15:56.380 epsilon regular. 01:15:56.380 --> 01:15:58.000 Then you're done. 01:15:58.000 --> 01:15:59.950 Whenever you run this loop, because we're 01:15:59.950 --> 01:16:02.080 doing the second step, your partition 01:16:02.080 --> 01:16:05.900 is always going to be equitable. 01:16:05.900 --> 01:16:09.110 But we now need to control the energy again 01:16:09.110 --> 01:16:12.260 to limit the number of steps. 01:16:12.260 --> 01:16:15.330 And the point here is that the first part still 01:16:15.330 --> 01:16:20.850 is exactly the same as before, where the energy goes up 01:16:20.850 --> 01:16:25.720 by at least epsilon to the minus 5. 01:16:25.720 --> 01:16:28.360 But the second part, the energy might go down, 01:16:28.360 --> 01:16:30.850 because we're no longer refining, just refining. 01:16:30.850 --> 01:16:34.270 Because we're doing some rebalancing. 01:16:34.270 --> 01:16:36.880 But you can do it in such a way that the amount of rebalancing 01:16:36.880 --> 01:16:39.220 that you do is really small. 01:16:39.220 --> 01:16:42.100 You're not actually changing the energy by so much. 01:16:42.100 --> 01:16:45.980 So I'll just hand wave here, and say 01:16:45.980 --> 01:16:52.980 that we can do this in such a way where the energy might 01:16:52.980 --> 01:16:56.550 go down, but only a little bit. 01:17:02.240 --> 01:17:04.040 So you're only changing a very small number 01:17:04.040 --> 01:17:07.430 of vertices, very small fraction of vertices. 01:17:07.430 --> 01:17:09.950 So if you change only an epsilon fraction of vertices, 01:17:09.950 --> 01:17:12.200 you don't expect the energy, which 01:17:12.200 --> 01:17:17.000 is something that comes out of summing pairs of vertex parts, 01:17:17.000 --> 01:17:18.290 to change by all that much. 01:17:21.540 --> 01:17:24.110 So putting these two together, you 01:17:24.110 --> 01:17:30.650 see that the energy still goes up by, let's say, at least 1/2 01:17:30.650 --> 01:17:35.780 of epsilon to the 5th power. 01:17:35.780 --> 01:17:39.800 And so then, the rest of the proof runs the same as before. 01:17:39.800 --> 01:17:42.590 You finish in some bounded number of steps. 01:17:42.590 --> 01:17:44.930 And you result in an equitable partition 01:17:44.930 --> 01:17:46.990 that's epsilon regular. 01:17:46.990 --> 01:17:48.790 I don't want to belabor the details. 01:17:48.790 --> 01:17:50.582 I mean, here, there's some things to check, 01:17:50.582 --> 01:17:52.960 but it's, I think, fairly routine. 01:17:52.960 --> 01:17:56.960 It's is more of an exercise in technical details. 01:17:56.960 --> 01:18:01.070 But the thing that actually is somewhat important 01:18:01.070 --> 01:18:04.277 is there's a wrong way to do this. 01:18:04.277 --> 01:18:06.360 I just want to point out that what's the wrong way 01:18:06.360 --> 01:18:09.610 to do this, is that you apply regularity lemma, 01:18:09.610 --> 01:18:11.150 and you think now it has something 01:18:11.150 --> 01:18:12.650 that's epsilon regular. 01:18:12.650 --> 01:18:18.350 Then I massage it to try to make it equitable at the end. 01:18:18.350 --> 01:18:21.040 And so if I don't look into the proof, 01:18:21.040 --> 01:18:23.500 I just look at a statement of Szemeredi's regularity lemma, 01:18:23.500 --> 01:18:25.490 and I get something that's epsilon regular, 01:18:25.490 --> 01:18:27.532 I say I'm just going to divide things up a little 01:18:27.532 --> 01:18:30.000 bit further, that doesn't work. 01:18:30.000 --> 01:18:32.280 Because the property of being epsilon 01:18:32.280 --> 01:18:37.340 regular is actually not preserved under refinement. 01:18:37.340 --> 01:18:38.560 So look at the definition. 01:18:38.560 --> 01:18:40.310 You have something that's epsilon regular. 01:18:40.310 --> 01:18:41.450 You refine the partition. 01:18:41.450 --> 01:18:45.960 If might fail to be epsilon regular. 01:18:45.960 --> 01:18:51.290 So you really have to take into the proof to get equitability. 01:18:51.290 --> 01:18:54.220 So just to repeat, a wrong way to try to get equitability 01:18:54.220 --> 01:18:56.930 is to apply regularity lemma, and at the end, 01:18:56.930 --> 01:18:59.420 try to massage it to get equitable. 01:18:59.420 --> 01:19:01.820 That doesn't work. 01:19:01.820 --> 01:19:05.690 Next time, I will show you how to apply Szemeredi's regularity 01:19:05.690 --> 01:19:07.240 lemma.