Ramsey numbers for set-colorings

For s, t, n ∈ N with s ≥ t, an (s, t)-coloring of K$_n$ is an edge coloring of Kn in which each edge is assigned a set of t colors from {1, . . . , s}. For k ∈ N, a monochromatic K$_k$ is a set of k vertices S such that for some color i ∈ [s], i ∈ c(uv) for all distinct u, v ∈ S. As in the case of the classical Ramsey number, we are interested in the least positive integer n = R$_{s,t}$(k) such that for any (s, t)-coloring of K$_n$, there exists a monochromatic K$_k$. We estimate upper and lower bounds for general cases and calculate close bounds for some small cases of R$_{s,t}$(k)

Stability for the Erdős-Rothschild problem

Given a sequence $\boldsymbol {k} := (k_1,\ldots ,k_s)$ of natural numbers and a graph G, let $F(G;\boldsymbol {k})$ denote the number of colourings of the edges of G with colours $1,\dots ,s$ , such that, for every $c \in \{1,\dots ,s\}$ , the edges of colour c contain no clique of order $k_c$ . Write $F(n;\boldsymbol {k})$ to denote the maximum of $F(G;\boldsymbol {k})$ over all graphs G on n vertices. This problem was first considered by Erdős and Rothschild in 1974, but it has been solved only for a very small number of nontrivial cases. In previous work with Pikhurko and Yilma, (Math. Proc. Cambridge Phil. Soc. 163 (2017), 341–356), we constructed a finite optimisation problem whose maximum is equal to the limit of $\log _2 F(n;\boldsymbol {k})/{n\choose 2}$ as n tends to infinity and proved a stability theorem for complete multipartite graphs G

On the structure of graphs with forbidden induced substructures

One of the central goals in extremal combinatorics is to understand how the global structure of a combinatorial object, e.g. a graph, hypergraph or set system, is affected by local constraints. In this thesis we are concerned with structural properties of graphs and hypergraphs which locally do not look like some type of forbidden induced pattern. Patterns can be single subgraphs, families of subgraphs, or in the multicolour version colourings or families of colourings of subgraphs. Erdős and Szekeres\u27s quantitative version of Ramsey\u27s theorem asserts that in every $2$-edge-colouring of the complete graph on $n$ vertices there is a monochromatic clique on at least $\frac{1}{2}\log n$ vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed colourings on subgraphs ensures much larger monochromatic cliques. The conjecture is open in general, though a few partial results are known. The first part of this thesis will be concerned with different variants of this conjecture: A bipartite variant, a multicolour variant, and an order-size variant for hypergraphs. In the second part of this thesis we focus more on order-size pairs; an order-size pair $(n,e)$ is the family consisting of all graphs of order $n$ and size $e$, i.e. on $n$ vertices with $e$ edges. We consider order-size pairs in different settings: The graph setting, the bipartite setting and the hypergraph setting. In all these settings we investigate the existence of absolutely avoidable pairs, i.e. fixed pairs that are avoided by all order-size pairs with sufficiently large order, and also forcing densities of order-size pairs $(m,f)$, i.e. for $n$ approaching infinity, the limit superior of the fraction of all possible sizes $e$, such that the order-size pair $(n,e)$ does not avoid the pair $(m,f)$

On the Existence of Loose Cycle Tilings and Rainbow Cycles

abstract: Extremal graph theory results often provide minimum degree conditions which guarantee a copy of one graph exists within another. A perfect $F$-tiling of a graph $G$ is a collection $\mathcal{F}$ of subgraphs of $G$ such that every element of $\mathcal{F}$ is isomorphic to $F$ and such that every vertex in $G$ is in exactly one element of $\mathcal{F}$. Let $C^{3}_{t}$ denote the loose cycle on $t = 2s$ vertices, the $3$-uniform hypergraph obtained by replacing the edges $e = \{u, v\}$ of a graph cycle $C$ on $s$ vertices with edge triples $\{u, x_e, v\}$, where $x_e$ is uniquely assigned to $e$. This dissertation proves for even $t \geq 6$, that any sufficiently large $3$-uniform hypergraph $H$ on $n \in t \mathbb{Z}$ vertices with minimum $1$-degree \delta^1(H) \geq {n - 1 \choose 2} - {\Bsize \choose 2} + c(t,n) + 1, where $c(t,n) \in \{0, 1, 3\}$, contains a perfect $C^{3}_{t}$-tiling. The result is tight, generalizing previous results on $C^3_4$ by Han and Zhao. For an edge colored graph $G$, let the minimum color degree $\delta^c(G)$ be the minimum number of distinctly colored edges incident to a vertex. Call $G$ rainbow if every edge has a unique color. For $\ell \geq 5$, this dissertation proves that any sufficiently large edge colored graph $G$ on $n$ vertices with $\delta^c(G) \geq \frac{n + 1}{2}$ contains a rainbow cycle on $\ell$ vertices. The result is tight for odd $\ell$ and extends previous results for $\ell = 3$. In addition, for even $\ell \geq 4$, this dissertation proves that any sufficiently large edge colored graph $G$ on $n$ vertices with $\delta^c(G) \geq \frac{n + c(\ell)}{3}$, where $c(\ell) \in \{5, 7\}$, contains a rainbow cycle on $\ell$ vertices. The result is tight when $6 \nmid \ell$. As a related result, this dissertation proves for all $\ell \geq 4$, that any sufficiently large oriented graph $D$ on $n$ vertices with $\delta^+(D) \geq \frac{n + 1}{3}$ contains a directed cycle on $\ell$ vertices. This partially generalizes a result by Kelly, K\"uhn, and Osthus that uses minimum semidegree rather than minimum out degree.Dissertation/ThesisDoctoral Dissertation Mathematics 201