18.319 | Fall 2005 | Graduate

Geometric Combinatorics


1 Sylvester-Gallai Theorem, Scott’s Problem about the Number of Distinct Slopes in the Plane Problem set 1 out
2 Scott’s Problem about the Number of Distinct Directions in Three-space

3 Motzkin-Rabin Theorem on Monochromatic Lines

4 Szemerédi-Trotter Theorem (Two Equivalent Formulations), Crossing Lemma

5 Unit Distances, Unit Area Triangles, Beck’s Two Extremities Theorem, Weak Dirac Conjecture Problem set 1 due

Problem set 2 out

6 Crossing Lemma for Multigraphs, Minimum Number of Distinct Distances

7 Pach-Sharir Theorem on Incidences of Points and Combinatorial Curves

8 Various Crossing Numbers, Embedding Technique

9 More on Crossing Numbers

Sum vs. Product Sets

Problem set 2 due

Problem set 3 out

10 Lipton-Tarjan and Gazit-Miller Separator Theorems

11 Cutting Circles into Pseudo-segments, Lenses, Transversal and Packing Numbers, d-intervals

12 Intersection Reverse Sequences

13 Arrangements, Levels, Lower Envelopes, Davenport-Schinzel Sequences Problem set 3 due

Problem set 4 out

14 Davenport-Schinzel Sequences of Order 3

15 Complexity of the First k-levels, Clarkson-Shor Technique, Cutting Lemma

16 Proof of Cutting Lemma, Simplicial Partitions

17 Spanning Trees with Low Stabbing Numbers Problem set 4 due

Problem set 5 out

18 k-levels, k-sets, Halving Lines

19 VC-dimension and ε-nets

20 Optimal ε-approximations, Links to Discrepancy Theory

21 Binary Space Partitions Problem set 5 due

Problem set 6 out

22 Geometric Graphs and Forbidden Subgraphs

23 Zig-zag and Alternating Paths for Disjoint Segments

24 Crossing-free Hamiltonian Cycles and Long Monochromatic Paths Problem set 6 due
25 Pseudo-triangulations and Art Gallery Problems

Course Info

As Taught In
Fall 2005