Problems in Ramsey theory, probabilistic combinatorics and extremal graph theory

Abstract

In this dissertation, we treat several problems in Ramsey theory, probabilistic combinatorics and extremal graph theory. We begin with the Ramsey theoretic problem of finding exactly m-coloured graphs. For which natural numbers m ∈ N are we guaranteed to find an m-coloured complete subgraph in any edge colouring of the complete graph on N? We resolve this question completely and prove, answering a question of Stacey and Weidl [104], that whenever we colour N(2) with infinitely many colours, we are guaranteed to find an ( n 2 ) -coloured complete subgraph for each n ∈ N. In addition, we also demonstrate that given a colouring of N(2) with k colours, there are at least √ 2k distinct values m ∈ [k] for which an infinite m-coloured complete subgraph exists. Finally, we also prove that given a colouring of N(2) with k colours and m ∈ [k], we can always find an infinite m-coloured complete subgraph for some m ∈ [k] such that |m− m| ≤ √ m/2. Next, we give some results in probabilistic combinatorics. First, we investigate the stability of the Erdős–Ko–Rado Theorem. For natural numbers n, r ∈ N with n ≥ r, the Kneser graph K(n, r) is the graph on the family of r-element subsets of {1, . . . , n} in which two sets are adjacent if and only if they are disjoint. Delete the edges of K(n, r) with some probability, independently of each other: is the independence number of this random graph equal to the independence number of the Kneser graph itself? We shall answer this question affirmatively as long as r/n is bounded away from 1/2, even when the probability of retaining an edge of the Kneser graph is quite small; we also prove a much more precise result when r = o(n1/3). We then study a geometric bootstrap percolation model on the three dimensional grid [n]3 called line percolation. In line percolation with infection parameter r, infection spreads from a subset A ⊂ [n]3 of initially infected lattice points as follows: if there is an axis parallel line L with r or more infected lattice points on it, then every lattice point of [n]3 on L gets infected and we repeat this until the infection can no longer spread. Our main result is the determination the critical density of initially infected points at which percolation (infection of the entire grid) becomes likely. Finally, we present two results in extremal graph theory. First, we consider a graph partitioning problem. For a graph G, let f(G) be the largest integer k such that there are two vertex-disjoint subgraphs of G, each on k vertices, inducing the same number of edges. We prove that f(G) ≥ n/2 − o(n) for every graph G on n vertices, settling a conjecture of Caro and Yuster [36]. Finally, we study the problem of cops and robbers on the grid where the robber is allowed to move faster than the cops. We prove that when the speed of the robber is a sufficiently large constant, the number of cops needed to catch the robber on an n×n grid is exp(Ω(log n/ log log n)).