Extremal Graph Colouring and Tiling Problems Jan Corsten A thesis submitted for the degree of Doctor of Philosophy Department of Mathematics The London School of Economics and Political Science 9th July 2020 Declaration I certify that the thesis I have presented for examination for the PhD degree of the London School of Economics and Political Science is solely my own work, with the exceptions outlined below. The copyright of this thesis rests with the author. Quotation from it is permitted, provided that full acknowledgement is made. In accordance with the regulations, I have deposited an electronic copy of it in LSE Theses Online held by the British Library of Political and Economic Science and have granted permission for my thesis to be made available for public reference. Otherwise, this thesis may not be reproduced without my prior written consent. I warrant that this authorisation does not, to the best of my belief, infringe the rights of any third party. I declare that this thesis consists of 46960 words. Statement of co-authored work The contents of Chapter 2 are well-known results of various authors not including myself. I confirm that Chapter 3 contains the following joint work. Section 3.1 is based on [18], which is jointly co-authored with Sebastián Bustamante, Nóra Frankl, Alexey Pokrovskiy and Jozef Skokan. Section 3.2 is based on [31], which is jointly co-authored with Walner Mendonça. I confirm that Chapter 4 contains the following joint work. Section 4.1 is based on [30], which is jointly co-authored with Louis DeBiasio, Ander Lamaison and Richard Lang. Section 4.2 is based on [29], which is jointly co-authored with Louis DeBiasio and Paul McKenney. I confirm that Chapter 5 is based on [3], which is jointly co-authored with Peter Allen, Julia Böttcher, Ewan Davies, Matthew Jenssen, Patrick Morris, Barnaby Roberts and Jozef Skokan. 2 Abstract In this thesis, we study a variety of different extremal graph colouring and tiling problems in finite and infinite graphs. Confirming a conjecture of Gyárfás, we show that for all :,A 2 N there is a constant ¡ 0 such that the vertices of every A-edge-coloured complete :-uniform hypergraph can be partitioned into a collection of at most monochromatic tight cycles. We shall say that the family of tight cycles has finite A-colour tiling number. We further prove that, for all natural numbers :, ? and A, the family of ?-th powers of :-uniform tight cycles has finite A-colour tiling number. The case where : = 2 settles a problem of Elekes, Soukup, Soukup and Szentmiklóssy. We then show that for all natural numbers Δ,A, every family F = f1, 2,...g of graphs with E¹=º = = and Δ¹=º ≤ Δ for every = 2 N has finite A-colour tiling number. This makes progress on a conjecture of Grinshpun and Sárközy. We study Ramsey problems for infinite graphs and prove that in every 2-edge- colouring of , the countably infinite complete graph, there exists a monochromatic N p infinite path % such that + ¹%º has upper density at least ¹12 ¸ 8º/17 ≈ 0.87226 and further show that this is best possible. This settles a problem of Erdős and Galvin. We study similar problems for many other graphs including trees and graphs of bounded degree or degeneracy and prove analogues of many results concerning graphs with linear Ramsey number in finite Ramsey theory. We also study a different sort of tiling problem which combines classical problems from extremal and probabilistic graph theory, the Corrádi–Hajnal theorem and (a special case of) the Johansson–Kahn–Vu theorem. We prove that there is some constant ¡ 0 such that the following is true for every = 2 3N and every ? ≥ =−2/3¹log =º1/3. If is a graph on = vertices with minimum degree at least 2=/3, then ? (the random subgraph of obtained by keeping every edge independently with probability ?) contains a triangle tiling with high probability. 3 Acknowledgements Firstly, I would like to thank my supervisors Peter and Julia, and Jozef, who acted like a third supervisor. I would like to express my gratitude for their guidance throughout my PhD, for all the discussions (mathematical and non-mathematical), and for encouraging me to participate in workshops, conferences and summer schools around the world. I would like to thank my fellow PhD students and good friends Attila and Nóra for our collaboration and for their unlimited supply of fun puzzles. I would like also like to thank all other PhD students for making the PhD office a great work environment, and for all the fun times we had at LSE and around the world. I would like to thank Kate, Enfale, Becca, Sarah and Ed for their administrative support and creating such a nice atmosphere in the department. I would like to thank Jozef and Jan for keeping us well hydrated. Finally, I would like to thank my parents and my sister for their constant support throughout my life. 4 Contents 1 Introduction 8 1.1 Notation ................................ 10 1.1.1 Elementary Notation ..................... 10 1.1.2 O-Notation .......................... 11 1.1.3 Graphs ............................ 12 1.1.4 Hypergraphs ......................... 14 1.1.5 Edge Colourings and Ramsey Numbers ........... 17 1.1.6 Infinite Graph Theory .................... 17 1.1.7 Random Graphs ....................... 18 1.2 Graphs with Linear Ramsey Number ................. 20 1.2.1 Paths and Cycles ....................... 21 1.2.2 Trees ............................. 22 1.2.3 Graphs with Bounded Degree ................ 22 1.3 Monochromatic Graph Tiling Problems ............... 24 1.3.1 History ............................ 24 1.3.2 New Results ......................... 25 1.4 Ramsey Problems for Infinite Graphs ................. 31 1.4.1 Infinite Paths ......................... 31 1.4.2 Infinite Trees ......................... 33 1.4.3 Infinite Graphs with “Linear Ramsey Number” ....... 34 1.5 Robust Triangle Tilings in Random Graphs ............. 38 1.5.1 Shamir’s Problem ....................... 38 1.5.2 Robustness .......................... 39 Contents 2 Preliminaries 42 2.1 The Absorption Method ........................ 42 2.2 Graph Regularity ........................... 44 2.2.1 Definitions and Basic Properties ............... 44 2.2.2 The Regularity Lemma .................... 49 2.2.3 Finding Large Regular Cylinders ............... 51 2.2.4 The Blow-up Lemma ..................... 53 2.3 Hypergraph Regularity ........................ 55 2.4 Probabilistic Tools .......................... 59 2.5 Entropy ................................ 60 3 Monochromatic Graph Tiling Problems 67 3.1 Tiling Coloured Hypergraphs with Tight Cycles ........... 67 3.1.1 Overview ........................... 67 3.1.2 Absorption Method for Hypergraphs ............. 68 3.1.3 Absorption Lemma ...................... 70 3.1.4 Proof of Theorem 1.3.10. ................... 75 3.2 Tiling Coloured Graphs with Graphs of Bounded Degree ...... 78 3.2.1 Overview ........................... 78 3.2.2 Tools for the Absorption Method ............... 80 3.2.3 The Absorption Lemma ................... 81 3.2.4 Proof of Theorem 1.3.16 ................... 88 3.2.5 Concluding Remarks ..................... 93 4 Ramsey Problems for Infinite Graphs 95 4.1 Ramsey Upper Density of Paths ................... 95 4.1.1 Overview ........................... 95 4.1.2 Proof of Theorem 1.4.4 .................... 96 4.1.3 Proof of Theorem 1.4.5 .................... 99 4.1.4 From Path Forests to Paths .................. 100 4.1.5 From Simple Forests to Path Forests ............. 102 4.1.6 Upper Density of Simple Forests ............... 105 4.1.7 Sequences and Oscillation .................. 110 6 Contents 4.2 Upper Density of Monochromatic Subgraphs ............ 114 4.2.1 Overview ........................... 114 4.2.2 Preliminaries ......................... 116 4.2.3 Constructions ......................... 118 4.2.4 Ultrafilters and Embedding .................. 121 4.2.5 Bipartite Ramsey Densities .................. 124 4.2.6 Trees ............................. 126 4.2.7 Graphs of Bounded Chromatic Number ........... 131 4.2.8 Graphs of Bounded Ruling Number ............. 133 5 Robust Triangle Tilings 136 5.1 Overview ............................... 136 5.1.1 Set-up ............................. 136 5.1.2 Proof Outline ......................... 139 5.2 Counting Triangles in Γ? ....................... 140 5.3 Embedding (Partial) Triangle Tilings ................. 147 5.3.1 Counting Almost Triangle Tilings .............. 147 5.3.2 Extending Almost Triangle Tilings .............. 148 5.3.3 Completing Triangle Tilings ................. 153 5.4 Proof of the Local Distribution Lemma ................ 153 5.4.1 A Simplification ....................... 153 5.4.2 Entropy lemma ........................ 155 5.4.3 Counting via Comparison .................. 165 5.5 Reducing to Regular Triples ..................... 169 5.5.1 Preparation .......................... 169 5.5.2 Reduction ........................... 179 7 1 Introduction Extremal graph theory seeks to answer questions of the following form: what is the largest or smallest value of some graph parameter among all graphs of a given class? The possibly first result in this area determines how many edges a triangle-free graph can have. Theorem 1.0.1 (Mantel [94]). Every graph with = vertices and more than =2/4 edges contains a triangle. Furthermore, the complete bipartite graph with parts of size d=/2e and b=/2c contains b=2/4c edges and is triangle-free. Another classical example of extremal graph theory is the study of Ramsey num- bers. The Ramsey number of a graph , denoted by R¹º,