Midterm Exam for First-Year Interest Group Seminar | N 1, Exams of Health sciences

Download Midterm Exam for First-Year Interest Group Seminar | N 1 and more Exams Health sciences in PDF only on Docsity! CS388C: Combinatorics and Graph Theory October 25, 2005 David Zuckerman Midterm Instructions. There are five problems of equal weight. You may assume any theorems stated in class or in the book, unless the question is to prove such a theorem. Explain all answers. You are allowed one sheet of paper, with writing on both sides. You have 75 minutes. Name: 1. Show that any family of more than 2n−1 subsets of [n] contains distinct sets A, B with A ⊂ B. 4. Show that there exists a constant c with the following property. Let M be an n×n matrix with ±1 entries. Then there exists a vector v with ±1 entries such that at least 99% of the entries in Mv are at most c √ n in absolute value. 5. Let f : Z × Z → Z be any symmetric function (f(x,y)=f(y,x)), and S be any set of 17 integers. Show that there exist distinct s, t, u ∈ S such that f(s, t) ≡ f(t, u) ≡ f(s, u) (mod 3).