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Notes By: Andrew Lin The Probabilistic Method in Combinatorics Lecturer: Professor Yufei Zhao Notes by: Andrew Lin Spring 2019 This is an edited transcript of the lectures of MIT’s Spring 2019 class 18.218: The Probabilistic Method in Combinatorics, taught by Professor Yufei Zhao. Each section focuses on a different technique, along with examples of applications. Additional course material, including problem sets, can be found on the course website. The main reference for the material is the excellent textbook N. Alon and J. H. Spencer, The probabilistic method, Wiley, 4ed. Most of the course will follow the textbook, though some parts will differ. Please contact Yufei Zhao ([email protected]) and Andrew Lin ([email protected]) for any questions or comments regarding these notes. Special thanks to Abhijit Mudigonda, Mihir Singhal, Andrew Gu, and others for their help in proofreading. Contents 1 Introduction to the probabilistic method 4 1.1 The Ramsey numbers ........................................... 4 1.2 Alterations ................................................. 5 1.3 Lovász Local Lemma ........................................... 6 1.4 Set systems ................................................ 7 1.5 Hypergraph colorings ........................................... 9 2 Linearity of expectation 11 2.1 Setup and basic examples ......................................... 11 2.2 Sum-free sets ............................................... 11 2.3 Cliques ................................................... 12 2.4 Independent sets ............................................. 13 2.5 Crossing numbers ............................................. 14 2.6 Application to incidence geometry .................................... 16 2.7 Derandomization: balancing vectors ................................... 18 2.8 Unbalancing lights ............................................. 19 2.9 2-colorings of a hypergraph ........................................ 19 2.10 High-dimensional sphere packings .................................... 21 1 3 Alterations 24 3.1 Dominating sets .............................................. 24 3.2 A problem from discrete geometry .................................... 25 3.3 Hard-to-color graphs ........................................... 26 3.4 Coloring edges ............................................... 27 4 The Second Moment Method 29 4.1 Refresher on statistics and concentration ................................ 29 4.2 Threshold functions for subgraphs .................................... 31 4.3 Clique number ............................................... 34 4.4 Chromatic number ............................................ 35 4.5 Number theory .............................................. 36 4.6 Distinct sums ............................................... 39 4.7 An application to analysis ......................................... 40 5 The Chernoff bound 41 5.1 Setup and proof .............................................. 41 5.2 An application: discrepancy ........................................ 42 5.3 Chromatic number and graph minors ................................... 43 6 The Lovász local lemma 45 6.1 Coloring: hypergraphs and real numbers ................................. 45 3 6.2 Coverings of R .............................................. 47 6.3 The general local lemma and proof .................................... 48 6.4 The Moser-Tardos algorithm ....................................... 50 6.5 A computationally hard example ..................................... 53 6.6 Back to independent sets ......................................... 53 6.7 Graphs containing large cycles ...................................... 54 6.8 Bounds on the linear arboricity conjecture ................................ 56 6.9 The lopsided local lemma ......................................... 59 6.10 Latin squares ............................................... 60 7 Correlation and Janson’s inequalities 62 7.1 The Harris-FKG inequality ........................................ 62 7.2 Applications of correlation ........................................ 63 7.3 The first Janson inequality: probability of non-existence ......................... 65 7.4 The second Janson inequality ....................................... 67 7.5 Lower tails: the third Janson inequality ................................. 69 7.6 Revisiting clique numbers ......................................... 71 7.7 Revisiting chromatic numbers ....................................... 72 8 Martingale convergence and Azuma’s inequality 74 8.1 Setup: what is a martingale? ....................................... 74 8.2 Azuma’s inequality ............................................ 75 8.3 Basic applications of this inequality .................................... 76 2 8.4 Concentration of the chromatic number ................................. 78 8.5 Four-point concentration? ........................................ 79 8.6 Revisiting an earlier chromatic number lemma .............................. 80 9 Concentration of measure 82 9.1 The geometric picture .......................................... 82 9.2 Results about concentration: median versus mean ............................ 83 9.3 High-dimensional spheres ......................................... 85 9.4 Projections onto subspaces ........................................ 87 9.5 What if we need stronger concentration? ................................ 88 9.6 Talagrand’s inequality: special case .................................... 88 9.7 Random matrices ............................................. 90 9.8 Talagrand’s inequality in general ..................................... 91 9.9 Increasing subsequences .......................................... 93 10 Entropy methods 96 10.1 Information entropy ............................................ 96 10.2 Various direct applications ........................................ 98 10.3 Bregman’s theorem ............................................ 99 10.4 A useful entropy lemma .......................................... 101 10.5 Entropy in graph theory .......................................... 103 10.6 More on graph homomorphisms: Sidorenko’s conjecture ........................ 107 11 The occupancy method 110 11.1 Introducing the technique ......................................... 110 11.2 An alternative approach to the above problem .............................. 113 11.3 Further bounds with the occupancy method ............................... 114 11.4 A useful corollary: Ramsey numbers ................................... 115 11.5 Back to independent sets ......................................... 116 11.6 Proper colorings in graphs ........................................ 117 12 A teaser for “Graph Theory and Additive Combinatorics” 118 12.1 A glance at Fermat’s last theorem .................................... 118 12.2 Turán’s theorem and more ........................................ 119 12.3 A generalization: more modern approaches ............................... 120 12.4 A principle about approaching complicated problems .......................... 120 12.5 Graph limits ................................................ 121 12.6 A few open problems ........................................... 122 3 Definition 0.1 (Asymptotic notation) Given functions or sequences f ; g > 0 (usually of some parameter n ! 1), the notation in each bullet point below are considered equivalent: • f . g; f = O(g); g = Ω(f ), f ≤ Cg (for some constant C); • f f g; f = o(g); g ! 0; g = !(f ). • f g; f = Θ(g); g . f . g. • f f ∼ g; g ! 1; f = (1 + o(1))g. Some event holds with high probability if its probability is 1 − o(1). Warning: analytic number theorists like to use the Vinogradov notation, where f g means f = O(g) instead of f = o(g) as we do. In particular, 100 1 is correct in Vinogradov notation. 1 Introduction to the probabilistic method In combinatorics and other fields of math, we often wish to show existence of some mathematical object. One clever way to do this is to try to construct this object randomly and then show that we succeed with positive probability. Proposition 1.1 jEj Every edge G = (V; E) with vertices V and edges E contains a bipartite subgraph with at least 2 edges. Proof. We can form a bipartite graph by partitioning the vertices into two groups. Randomly color each vertex either white or black (making the white and black sets the two groups), and include only the edges between a white and a black edge in a new graph G0. Since all vertices are colored independently at random, each edge is included in G0 with 1 jEj probability 2 . Thus, we have an average of 2 edges in our graph by linearity of expectation, and this means that at least one coloring will work. This class will introduce a variety of methods to solve these types of problems, and we’ll start with a survey of those techniques. 1.1 The Ramsey numbers Definition 1.2 Let the Ramsey number R(k; `) be the smallest n such that if we color the edges of Kn (the complete graph on n vertices) red or blue, we always have a Kk that is all red or a K` that is all blue. Theorem 1.3 (Ramsey, 1929) For any integers k; `, R(k; `) is finite. One way to do this is to use the recurrence inequality R(r; s) ≤ R(r − 1; s) + R(r; s − 1) by picking an arbitrary vertex v and partitioning the remaining vertices by the color of their edge to v. 4 Theorem 1.4 (Erdős, 1947) We have R(k; k) > n for all n 1−(k ) 2 2 < 1: k In other words, for any n that satisfies this inequality, we can color Kn with no monochromatic Kk . Proof. Color the edges of Kn randomly. Given
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