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Probabilistic Methods in Combinatorics

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Probabilistic Methods in Combinatorics Probabilistic Methods in Combinatorics Joshua Erde Department of Mathematics, Universit¨atHamburg. Contents 1 Preliminaries 5 1.1 Probability Theory . .5 1.2 Useful Estimates . .8 2 The Probabilistic Method 10 2.1 Ramsey Numbers . 10 2.2 Set Systems . 11 3 The First Moment Method 15 3.1 Hamiltonian Paths in a Tournament . 16 3.2 Tur´an'sTheorem . 17 3.3 Crossing Number of Graphs . 18 4 Alterations 20 4.1 Ramsey Numbers Again . 20 4.2 Graphs of High Girth and High Chromatic Numbers . 21 5 Dependent random choice 24 5.1 Tur´anNumbers of Bipartite Graphs . 25 1 5.2 The Ramsey Number of the Cube . 26 5.3 Improvements . 27 6 The Second Moment Method 29 6.1 Variance and Chebyshev's Inequality . 29 6.2 Threshold Functions . 30 6.3 Balanced Subgraphs . 33 7 The Hamiltonicity Threshold 35 7.1 The Connectivity Threshold . 35 7.2 Pos´a'sRotation-Extension Technique . 39 7.3 Hamiltonicty Threshold . 42 8 Strong Concentration 44 8.1 Motivation . 44 8.2 The Chernoff Bound . 44 8.3 Combinatorial Discrepancy . 48 8.4 A Lower Bound for the Binomial Distribution . 49 9 The Lov´asLocal Lemma 52 9.1 The Local Lemma . 52 9.2 Ramsey Bounds for the last time . 55 9.3 Directed Cycles . 56 9.4 The Linear Aboricity of Graphs . 57 10 Martingales and Strong Concentration 62 10.1 The Azuma-Hoeffding Inequality . 62 2 10.2 The Chromatic Number of a Dense Random Graph . 66 10.3 The Chromatic Number of Sparse Random Graphs . 70 11 Talagrand's Inequality 73 11.1 Longest Increasing Subsequence . 75 11.2 Chromatic Number of Graph Powers . 76 11.3 Exceptional outcomes . 80 12 Entropy Methods 86 12.1 Basic Results . 86 12.2 Br´egman'sTheorem . 88 12.3 Shearer's lemma and the Box theorem . 91 12.4 Independent Sets in a Regular Bipartite Graph . 95 12.5 Bipartite Double Cover . 96 13 Derandomization and Combinatorial Games 98 13.1 Maximum Cuts in Graphs . 98 13.2 Ramsey graphs . 99 13.3 Positional Games . 100 13.4 Weak Games . 105 13.5 The Neighbourhood Conjecture . 107 14 The Algorithmic Local Lemma 111 3 Preface These notes were used to lecture a course at UHH for masters level students in the winter semester of 2015, and again in the Summer Semester of 2018. A large part of the course material was heavily based on the excellent set of lecture notes \The Probabilistic Method" by Matouˇsek and Vondr´ak,which themselves follow the book of the same name by Alon and Spencer. The latter cannot be recommended highly enough as an introductory text to the subject and covers most of the material in these lectures (and much more) in depth. There are a few reasons for writing yet another set of notes on the same subject. Firstly there is simply the matter of length. German semesters are slightly longer than some other university terms, and so the lecture course was to run for 26 lectures of 90 minutes each, which would easily have exhausted the material in the previous notes with plenty of time to spare. This extra time has allowed us to include more examples of applications of the method, such as the beautiful result on the threshold for Hamiltonicity in Section 7, as well as chapters on wider topics not usually covered in an introductory course. Secondly the course was designed with graph theorists in mind, with examples chosen as much as possible with an emphasis on graph theoretical problems. This was not always possible, and indeed in some cases not always preferable as there are many beautiful and instructive examples arising naturally from set systems or hypergraphs, but hopefully the result was that the students were able to easily understand the statement of the results and motivations for them. The material from Section 5 on the method of dependent random choice comes from the survey paper \Dependent Random Choice" by Fox and Sudakov, which is a good introductory text if you wish to know more about the technique. Similarly the material from Section 15 on pseudorandomness follows in part the survey paper \Pseudo-random graphs" by Krivelevich and Sudakov. Section 11.3 comes from the paper \A stronger bound for the strong chromatic index" by Bruhn and Joos and Section 14 follows in part a lecture of Alistair Sinclair. 4 1 Preliminaries 1.1 Probability Theory This section is intended as a short introduction to the very basics of probability theory, covering only the basic facts about finite probability spaces that we will need to use in this course. Ω Definition. A probability space is a triple (Ω; Σ; P), where Ω is a set, Σ ⊆ 2 is a σ-algebra (A non-empty collection of subsets of Ω which is closed under taking complements and countable unions/intersections), and P is a measure on Σ with P(Ω) = 1. That means: • P is non-negative; • P(;) = 0; n • For all countable collections of disjoint sets fAigi=1 in Σ, 1 ! 1 [ X P Ai = P(Ai): i=1 i=1 The elements of Σ are called events and the elements of Ω are called elementary events. For an event A, P(A) is called the probability of A. During this course we will mostly consider finite probability spaces, those where Ω is finite Ω and Σ = 2 . In this case the probability measure P is determined by the value it takes on P elementary events. That is, given any function p :Ω ! [0; 1] that satisfies !2Ω p(!) = 1, then P the function on Σ given by P(A) = !2A p(!) is a probability measure. In a finite probability space, the most basic example of a probability measure is the uniform distribution on Ω, where jAj (A) = for all A ⊆ Ω: P jΩj An important example of a probability space, which we will return to throughout the course, is that of a random graph. Definition. The probability space of random graphs G(n; p) is a finite probability space whose elementary events are all graphs on a fixed set of n vertices, and where the probability of each graph with m edges is m n −m p(G) = p (1 − p)(2) This corresponds to the usual notion of picking a random graph by including every potential edge independently with probability p. We will often denote an arbitary event from this space by G(n; p). We note that p can, and often will be, a function of n. Given a property of graphs P we say that G(n; p) satisfies P almost surely or with high probability if lim (G(n; p) satisfies P ) = 1: n!1 P 5 Another natural, and often utilized, probability space related to graphs is that of G(n; M) where the elementary events are all graphs on n vertices with M edges, with the uniform probability measure. G(n; M) is, except in certain cases, much less nice to work with than G(n; p). We won't work with this space in the course, and mention that, quite often in applications you can avoid working with G(n; M) at all by proving the result you want in G(n; p) for an appropriate p and using some standard theorems that translate results between the spaces. One elementary fact that we will use often is the following, often referred to as the union bound: Lemma 1.1 (Union bound). For any A1;A2;:::;An 2 Σ, n ! [ n P Ai ≤ Σi=1P(Ai) i=1 Proof. For each 1 ≤ i ≤ n let us define Bi = Ai n (A1 [ A2 [ ::: [ Ai−1): S S Then Bi ⊂ Ai, and so P(Bi) ≤ P(Ai), and also Bi = Ai. Therefore, since the events B1;B2;:::;Bn are disjoint, by the additivity of P n ! n ! [ [ n n P Ai = P Bi = Σi=1P(Bi) ≤ Σi=1P(Ai) i=1 i=1 Definition. Two events A; B 2 Σ are independent if P(A \ B) = P(A)P(B): More generally, a set of events fA1;A2;:::;Ang is mutually independent if, for any subset of indices I ⊆ [n], ! \ Y P Ai = P(Ai): i2I i2I It is important to note that the notion of mutual independence is stronger than simply having pairwise independence of all the pairs Ai;Aj. Intuitively, the property of independence of two events, A and B, should mean that knowledge about whether or not A occurs should not influence the likelihood of B occurring. This intuition is made formal with the idea of conditional probability. Definition. Given two events A; B 2 Σ such that P(B) 6= 0, we define the conditional probability of A, given that B occurs, as P(A \ B) P(AjB) = : P(B) Note that, as expected, A and B are independent if and only if P(AjB) = P(A). Definition. A real random variable on probability space (Ω; Σ; P) is a function X :Ω ! R that is P-measurable. (That is, for any a 2 R; f! 2 Ω: X(!) ≤ ag 2 Σ.) 6 For the most part all the random variables we consider will be real valued, so for convenience throughout the course we will suppress the prefix "real". However we will in some contexts later in the course consider random variables which are functions from Ω to more general spaces. In the case of a finite probability space, any function X :Ω ! R will define a random variable. Given a measurable set A ⊆ R the probability that the value X takes lies in A is P(f! 2 Ω: X(!) 2 Ag) which we will write as P(X 2 A).
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