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MAT 307 - Spring 2009 Assignment 4 Due: April 15 The solution for each problem should be no longer than one page. Problem 1. [4 points] Let G1 and G2 be two graphs on the same vertex set V . Prove that the chromatic number of their union G1 ∪G2 (we take the union of the edge sets of both graphs) satisfies χ(G1 ∪ G2 ) ≤ χ(G1 ) · χ(G2 ). Use this to show that if H1 , . . . , Ht are bipartite graphs whose union is a complete graph on n vertices, then t ≥ log2 n. Problem 2. [6 points] Let k be a positive integer. Suppose T is a tournament with at least k vertices such that every subset U of k vertices from T is dominated by some vertex v 6∈ U , i.e., v −→ u for all u ∈ U . We have seen in class that there are such tournaments with O(k 2 2k ) vertices. Prove that there is no such tournament with 2k vertices. Problem 3. [6 points] Let A1 , . . . , Am be subsets of an n-element sets such that |Ai | is not divisible by 6 for every i, but the sizes of all pairwise intersections |Ai ∩ Aj | are divisible by 6. Prove that m ≤ 2n. Problem 4. [6 points] Prove that every three-uniform hypergraph with n vertices and m ≥ n/3 hy2n3/2 √ . peredges contains an independent set of size at least 3√ 3 m Problem 5. [8 points] Let F be a finite collection of binary strings of finite length and assume no member of F is a prefix of another one. Let Ni denote the number of strings of length i in F . Prove that X Ni ≤ 1. 2i i 1