Extremal graph colouring and tiling problems

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 k, r ∈ N there is a constant C > 0 such that the vertices of every r-edge-coloured complete k-uniform hypergraph can be partitioned into a collection of at most C monochromatic tight cycles. We shall say that the family of tight cycles has finite r-colour tiling number. We further prove that, for all natural numbers k, p and r, the family of p-th powers of k-uniform tight cycles has finite r-colour tiling number. The case where k = 2 settles a problem of Elekes, Soukup, Soukup and Szentmiklóssy. We then show that for all natural numbers ∆, r, every family F = {F1, F2, . . .} of graphs with v (Fn) = n and ∆(Fn) ≤ ∆ for every n ∈ N has finite r-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 KN, the countably infinite complete graph, there exists a monochromatic infinite path P such that V (P) has upper density at least (12 + √8)/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 C > 0 such that the following is true for every n ∈ 3N and every p ≥ Cn−2/3 (log n)1/3. If G is a graph on n vertices with minimum degree at least 2n/3, then Gp (the random subgraph of G obtained by keeping every edge independently with probability p) contains a triangle tiling with high probability

Ramsey numbers of hypergraphs of a given size

The $q$-color Ramsey number of a $k$-uniform hypergraph $H$ is the minimum integer $N$ such that any $q$-coloring of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. The study of these numbers is one of the central topics in Combinatorics. In 1973, Erd\H{o}s and Graham asked to maximize the Ramsey number of a graph as a function of the number of its edges. Motivated by this problem, we study the analogous question for hypergaphs. For fixed $k \ge 3$ and $q \ge 2$ we prove that the largest possible $q$-color Ramsey number of a $k$-uniform hypergraph with $m$ edges is at most $\mathrm{tw}_k(O(\sqrt{m})),$ where $\mathrm{tw}$ denotes the tower function. We also present a construction showing that this bound is tight for $q \ge 4$. This resolves a problem by Conlon, Fox and Sudakov. They previously proved the upper bound for $k \geq 4$ and the lower bound for $k=3$. Although in the graph case the tightness follows simply by considering a clique of appropriate size, for higher uniformities the construction is rather involved and is obtained by using paths in expander graphs

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 $(m,f)$

Results on the Generalised Shift Graph

In the paper ‘On Chromatic Number of Infinite Graphs’ (1968), Erdős and Hajnal defined the Shift Graph to be the graph whose vertices are the n-element subsets of some totally ordered set S, regarded as increasing n-tuples, such that A = (a1, ..., an) and B = (b1, ..., bn) are neighbours iff a1 < b1 = a2 <b2 = a3 < ... < bn−1 = an < bn or the other way round. In the paper ‘On Generalised Shift Graphs’ (2014), Avart, Łuczac and Rödl extend this definition to include all possible arrangements of the ais and bis, known as ‘types’. In this thesis, we will consider a selection of these types and study the corresponding graphs. All the types we consider will be written as 1^k3^m2^k, where k + m = n, which means that the final m entries of (a1, ..., an) are identified with the first m entries of (b1, ..., bn). Such a graph with totally ordered set S and type 1^k3^m2^k is denoted G(S,1^k3^m2^k). There are two related questions here. One is when the (undirected) graphs G(S,1^k3^m2^k) and G(S',1^k3^m2^k) are distinct (non-isomorphic) for distinct linear orderings S, S'. The other is to what extent we can recognise S inside the graph (called ‘reconstruction’). A positive solution to the latter also yields one for the former, since if we can recognise S in its graph, and S′ in its graph, and they are distinct, then so must the graphs be. We focus on these main cases: S is finite, S is an ordinal, S is a more general totally ordered set. The tools available for reconstruction depend on whether S is a total ordering, a dense total ordering, or an ordinal. There are additional technical complications in the case where S has endpoints, and similarly for S containing relatively small finite segments. Since these graphs are undirected, we expect in general only to recover a linear ordering up to order reversal. The natural notion here is of ‘linear betweenness’, and we spend some time studying linear betweenness relations in their own right, also considering the induced relations on n-tuples. Betweenness relations on n-tuples correspond to shift graphs of the special form G(S,1^n2^n) (i.e. in which no identifications are made). The main contribution of the thesis is to show how it is possible in many instances to reconstruct the underlying linear order (often just up to order-reversal) from the generalized shift graph. A typical example of this is Theorem 4.4. The techniques are to employ graph-theoretical features of the relevant shift graph, such as co-cliques or pairs of co-cliques fulfilling various conditions to ‘recognize’ points and relations of the underlying linear order. There are many variants depending on the precise circumstances (dense or not, with or without endpoints, well-ordered, only partially ordered). We show that for ordinals α and β, if G(α,1^k3^m2^k) is isomorphic to G(β,1^k3^m2^k) then α = β. Note that the fact that (in the infinite case) α is not isomorphic to its reversed ordering means that the betweenness relation is enough to give us the ordering. This result does not necessarily extend to all total orderings in full generality, but we obtain many results. A suite of techniques is used, which may be adapted suitably depending on circumstances, endpoints or not, density, or finiteness. In a more open-ended chapter, we generalise as much of the material for total orders to partial orders, the easiest case being that of trees. Work by Rubin [15] considers reconstruction in a slightly different sense: that a structure can be reconstructed from its automorphism group. So we have two ‘levels’ of reconstruction: of the graph from its automorphism group, and then if possible of the underlying total order from the graph. With this in mind, we study the automorphism groups of many of the graphs arising, managing in several cases to give quite explicit descriptions, so answering Rubin’s reconstruction question - i.e. whether or not a structure can be ‘re- constructed’ from its automorphism group (as in for example [17]) - where possible. For instance, we show that it is possible to determine S from Aut(G(S,132)) if and only if G(S, 132) contains no two points sharing exactly the same neighbour sets. Finally we return to colouring questions as in the original paper of Erdős and Hajnal, and show that the chromatic number of G(κ, 132) is equal to κ for any strong limit cardinal κ

Discrete Geometry and Convexity in Honour of Imre Bárány

This special volume is contributed by the speakers of the Discrete Geometry and Convexity conference, held in Budapest, June 19–23, 2017. The aim of the conference is to celebrate the 70th birthday and the scientific achievements of professor Imre Bárány, a pioneering researcher of discrete and convex geometry, topological methods, and combinatorics. The extended abstracts presented here are written by prominent mathematicians whose work has special connections to that of professor Bárány. Topics that are covered include: discrete and combinatorial geometry, convex geometry and general convexity, topological and combinatorial methods. The research papers are presented here in two sections. After this preface and a short overview of Imre Bárány’s works, the main part consists of 20 short but very high level surveys and/or original results (at least an extended abstract of them) by the invited speakers. Then in the second part there are 13 short summaries of further contributed talks. We would like to dedicate this volume to Imre, our great teacher, inspiring colleague, and warm-hearted friend

Decomposition of Geometric Set Systems and Graphs

We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold covering of the plane with its translates is decomposable into two disjoint coverings of the whole plane. Pach conjectured that every convex set is cover-decomposable. We verify his conjecture for polygons. Moreover, if m is large enough, we prove that any m-fold covering can even be decomposed into k coverings. Then we show that the situation is exactly the opposite in 3 dimensions, for any polyhedron and any $m$ we construct an m-fold covering of the space that is not decomposable. We also give constructions that show that concave polygons are usually not cover-decomposable. We start the first part with a detailed survey of all results on the cover-decomposability of polygons. The second part investigates another geometric partition problem, related to planar representation of graphs. The slope number of a graph G is the smallest number s with the property that G has a straight-line drawing with edges of at most s distinct slopes and with no bends. We examine the slope number of bounded degree graphs. Our main results are that if the maximum degree is at least 5, then the slope number tends to infinity as the number of vertices grows but every graph with maximum degree at most 3 can be embedded with only five slopes. We also prove that such an embedding exists for the related notion called slope parameter. Finally, we study the planar slope number, defined only for planar graphs as the smallest number s with the property that the graph has a straight-line drawing in the plane without any crossings such that the edges are segments of only s distinct slopes. We show that the planar slope number of planar graphs with bounded degree is bounded.Comment: This is my PhD thesi