Most of the problems are assigned from the required textbook Bona, Miklos. *A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory*. World Scientific Publishing Company, 2011. ISBN: 9789814335232. [Preview with Google Books]

A problem marked by * is difficult; it is not necessary to solve such a problem to do well in the course.

## Problem Set 12

- Due in Session 37
- Practice Problems
- Session 33: Chapter 12: Exercises 1, 3, 4, 10
- Session 34: None
- Session 35: Chapter 12: Exercise 22
- Session 36: Chapter 12: Exercises 7, 8, 9
- Session 37: Chapter 13: Exercises 4, 6, 7, 8. By "the space," the author means three-dimensional Euclidean space. (It is common to say "the plane" for the Euclidean plane, but I never heard "the space" before.)
- Non-practice" problems, but not to be handed in: Chapter 13: Exercises 20, 24, 38 in 24, the phrase "five persons on the bus who know at least four of the other passengers" means "five persons on the bus,
*each*of whom know at least four of the other passengers". The definition of a permutation of an infinite set*S*in 38 is vague and incorrect. Regard a permutation of the positive integers as a sequence*a*_{1},*a*_{2}, ... of positive integers such that every positive integer occurs exactly once. We say that this sequence*contains*an increasing arithmetic progression of length three if there exist*i*<*j*<*k*such that*a*_{i},*a*_{j},*a*_{k}is an increasing arithmetic progression.

- Non-practice" problems, but not to be handed in: Chapter 13: Exercises 20, 24, 38 in 24, the phrase "five persons on the bus who know at least four of the other passengers" means "five persons on the bus,

- Problems Assigned in the Textbook
- Chapter 12: Exercises 13, 18, 19, 20, 21. For 20, note that it is o.k. for
*p*_{a}=0 or*p*_{b}=0.

- Chapter 12: Exercises 13, 18, 19, 20, 21. For 20, note that it is o.k. for
- Additional Problems
- (A19) Let
*n*≥4. Suppose that*P*is a convex polyhedron with one*n*-vertex face and two additional vertices. What is the most number of faces that*P*can have? - (A20) Show that the complete graph
*K*_{7}can be embedded on a torus without crossing edges. Draw the torus as a square (or rectangle) with opposite edges identified. - (A21*) Find a triangle-free graph (i.e., no three vertices that are pairwise adjacent) with chromatic number four. Try to use as few vertices as possible.

- (A19) Let
- Bonus Problems
- (B5) Find three states of the United States with the property that there exist three points in common to all three of them.
**Note**. Suppose we choose a point in the interior of each of the states and join it to the three intersection points. Why does this not give a planar embedding of*K*_{3},_{3}?

- (B5) Find three states of the United States with the property that there exist three points in common to all three of them.