Math 261A Probabilistic Combinatorics Instructor: Sam Buss Fall 2015 Homework assignments

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1 Math 261A Probabilistic Combinatorics Instructor: Sam Buss Fall 2015 Homework assignments Problems are mostly taken from the text, The Probabilistic Method (3rd edition) by N Alon and JH Spencer Please do approximately two problems per assignments (Assignments have been running about once every fourth lecture) Double-starred problems are research problems 1 Prove that if there is a real p, 0 p 1 such that ( ) ( ) n n p (k 2) + (1 p) (t 2) < 1, k t then R(k,t) > n From this, show that R(4,t) Ω(t 3/2 /(lnt) 3/2 ) 2 Let n 4 and H be an n-uniform hypergraph with at most 4 n 1 /3 n many edges Prove there exists a coloring of the vertices of H by four colors such that every edge has all four colors represented 3 (Kraft inequality for prefix free codes) Let F be a finite set of (finite) binary strings Assume no string in F is a prefix of any other string in F (So F is prefix-free ) Let N i be the number of strings in F of length i Prove that N i 2 i 1 i Prove this bound is tight by giving an example where equality holds 4 As discussed in class for the deterministic construction for the proof of Theorem122: At thetop of page6, thereis aboundr(δ+1)/n, which can be improved to r(δ +1)/(n r) Can this better bound be used to prove a stronger form of Theorem 122 with a better bound on the size of a dominating set G? If so, what better bound is achievable? 5 Let n 2, and H = (V,E) be an n-uniform hypergraph with E = 4 n 1 edges Prove there is a coloring of V with four colors so that no edge is monochromatic 1

2 6 Let X be a set of n non-zero real numbers (a) Prove that X has a sum-free subset A of cardinality n/3 (b) Prove that X has a sum-free subset A of cardinality > n/3 [Hint: Define Sign(x) to 0 or ±1 depending on the sign of x Let X = {x i } i Prove there are rational ( numbers c i so that for for ) ( ) all values ǫ i {0,1, 1}, Sign i ǫ ix i = Sign i ǫ ic i [Alon- Kleitman]] 7 Let G = (V,E) be a bipartite graph with n vertices Suppose each vertex v has a list S(v) of more than log 2 n many colors which are permitted be assigned to v Prove there is a proper coloring of the vertices of G which assigns to each vertex a color from its list S(v) 8 Suppose n 2k and k 2, and that F is a family of intersecting k- subsets of [n] Also suppose F = Prove that F k 2( n 2 k 2) [EKR, HM, ABBCI] 9 Proposition 111 showed that if ( n k) 2 1 ( k 2) < 1, then R(k,k) > n Show that this implies R(k,k) k e 2 2k/2 (1 o(1)) Theorem 311 showed that if R(k,k) > n ( n k) 2 1 ( k 2) Show that this implies R(k,k) k e 2k/2 (1 o(1)) Hint: Give a good upper bound on ( n k) Remark: In both cases, (1 o(1)) can be replaced by (1+o(1)) 10 Prove that every 3-uniform hypergraph with n vertices m n/3 edges contains an independent set (a set of vertices containing no edges) of size at least 2n 3/2 3 3m 11 (Kraft-McMillan inequality for uniquely decipherable codes) Let F be a finite set of finite binary strings, ie, F {0,1} Assume that no two distinct concatenations of finitely many strings from F result 2

3 in the same binary string That is, if v 1 v 2 v l = w 1 w 2 w k for v i,w i F, then l = k and each v i = w i Prove that i N i 2 i 1, where as in problem 3, N i is the number of strings in F of length i 12 Suppose p n > 10m 2 with p prime Let 0 < a 1 < a 2 < a m < p be integers Prove that there is an integer x for which the m numbers are distinct (xa i mod p) mod n 13 Let F be a family of subsets of [n] = {0,,n 1} Suppose also there are no A,B F with A a proper subset of B Let σ S n be a random permutation of [n], and consider the random variable X = {i : σ(0),,σ(i 1) F} By considering the expected value of X, prove that F ( n n/2 ) 14 The geometric distribution with parameter p (0, 1] is defined as follows Let a coin have probability p of heads A random integer X is selected by flipping the coin until a heads first occurs, and setting X equal to the number of coin flips (So X 1) a Prove that Pr[X = j] = (1 p) j 1 p Compare this to the power series for p 1 (1 p)t b Derive the expectation E[X] (Answer: 1/p) c Derive the variance Var[X] (Answer: (1 p)/p 2 ) You are encouraged to usemethods similar to what we usedin class for the Poisson distribution Hint: Work with Y = X 1; and differentiate the power series expression for p 1 (1 p)t (twice) 15 Let X be a random variable with expectation E[X] = 0 and variance σ 2 Prove that, for λ > 0, Pr[X λ] σ 2 σ 2 +λ 2 3

4 16 Let H be a graph Let H be a subgraph of H (not necessarily a proper subgraph) with v vertices and e edges which maximizes the density ρ(h ) = e /v Prove that r(n) = n v /e is the threshold function for the graph property H is a subgraph of G 17 From the proof of Theorem 451 (Wednesday 11/03 in class), let ( k n k ) g(i) = i)( k i ( n 2 k) (i 2) Prove that g(i) g(3)+g(k 1) for all 3 < i < k 1 [Bollobás-Erdős 76] 18 Let G = (V,E) be a simple graph, and suppose each vertex v Vis associated with a set of colors S(v) of cardinality at least 10d where d 1 Suppose also that, for each vertex x and each c S(v) there are at most d neighbors u of v such that c S(u) Prove there is a proper coloring of G assigning each vertex v a color from S(v) 19 Prove that, for every ǫ > 0, there is a finite l = l(ǫ) and there is an infinite sequence of bits a 1,a 2,a 3, such that for every i 1, the two bit vectors a i,a i+1,a i+l 1 and a i+l,a i+l+1,a i+2l 1 differ in at least ( 1 2 ǫ)l bits 20 Prove that, for every ǫ > 0, there is a finite l 0 = l 0 (ǫ) and there is an infinitesequence of bits a 1,a 2,a 3, such that for l l 0 and every i 1, thetwobitvectors a i,a i+1,a i+l 1 and a i+l,a i+l+1,a i+2l 1 differ in at least ( 1 2 ǫ)l bits 21 Let G = (V,E) be a cycle of length 4n = V Let V 1,,V n be a partition of V into sets of size 4 Prove or give a counterexample: There must an independent set of vertices of G of size n containing one vertex from each V i 22 Let F be a set of m open unit balls in R 3 Prove that the balls in F split R 3 into < m 3 many connected components 23 Extend the Moser-Tardős constructive proof to work with only the weak dependency condition (top of page 70) In particular, can this work with Theorem 561 on Latin traversals? 24 Let P is the probability that a random G = G(n,1/2) consists of a single connected component Let Q be the probability that when a graph on [n] has its edges colored red or blue independently with equal 4

5 probabilities, then both the red edges define a connected graph on [n] and the blue edges define a connected graph on [n] Prove or disprove: Q P 2 25 For i = 1,,k, let F i be an intersecting family of subsets of [n] Prove that k 2 n 2 n k i=1 F i 5