Ordered Ramsey numbers of loose paths and matchings

For a $k$-uniform hypergraph $G$ with vertex set $\{1,\ldots,n\}$, the ordered Ramsey number $\operatorname{OR}_t(G)$ is the least integer $N$ such that every $t$-coloring of the edges of the complete $k$-uniform graph on vertex set $\{1,\ldots,N\}$ contains a monochromatic copy of $G$ whose vertices follow the prescribed order. Due to this added order restriction, the ordered Ramsey numbers can be much larger than the usual graph Ramsey numbers. We determine that the ordered Ramsey numbers of loose paths under a monotone order grows as a tower of height one less than the maximum degree. We also extend theorems of Conlon, Fox, Lee, and Sudakov [Ordered Ramsey numbers, arXiv:1410.5292] on the ordered Ramsey numbers of 2-uniform matchings to provide upper bounds on the ordered Ramsey number of $k$-uniform matchings under certain orderings.Comment: 13 page

Polychromatic colorings of certain subgraphs of complete graphs and maximum densities of substructures of a hypercube

If G is a graph and H is a set of subgraphs of G, an edge-coloring of G is H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, polyHG, is the largest number of colors in an H-polychromatic coloring. We determine polyHG exactly when G is a complete graph on n vertices, q a fixed nonnegative integer, and H is the family of one of: all matchings spanning n-q vertices, all 2-regular graphs spanning at least n-q vertices, or all cycles of length precisely n-q. For H, K, subsets of the vertex set V(Qd) of the d-cube Qd, K is an exact copy of H if there is an automorphism of Qd sending H to K. For a positive integer, d, and a configuration in Qd, H, we define λ(H,d) as the limit as n goes to infinity of the maximum fraction, over all subsets S of V(Qn), of sub-d-cubes of Qn whose intersection with S is an exact copy of H. We determine λ(C8,4) and λ(P4,3) where C8 is a “perfect” 8-cycle in Q4 and P4 is a “perfect” path with 4 vertices in Q3, λ(H,d) for several configurations in Q2, Q3, and Q4, and an infinite family of configurations. Strong connections exist with extensions Ramsey numbers for cycles in a graph, counting sequences with certain properties, inducibility of graphs, and we determine the inducibility of two vertex disjoint edges in the family of bipartite graphs

Extremal results for berge hypergraphs

Let E(G) and V (G) denote the edge set and vertex set of a (hyper)graph G. Let G be a graph and H be a hypergraph. We say that a hypergraph H is a Berge-G if there is a bijection f : E(G) → E(H) such that for each e ϵ E(G) we have e ? f(e). This generalizes the established definitions of "Berge path" and "Berge cycle" to general graphs. For a fixed graph G we examine the maximum possible size of a hypergraph with no Berge-G as a subhypergraph. In the present paper we prove general bounds for this maximum when G is an arbitrary graph. We also consider the specific case when G is a complete bipartite graph and prove an analogue of the K?ovári-Sós-Turán theorem. In case G is C4, we improve the bounds given by Gy?ori and Lemons [Discrete Math., 312, (2012), pp. 1518-1520]. © 2017 Society for Industrial and Applied Mathematics

Extremal results in hypergraph theory via the absorption method

The so-called "absorbing method" was first introduced in a systematic way by Rödl, Ruciński and Szemerédi in 2006, and has found many uses ever since. Speaking in a general sense, it is useful for finding spanning substructures of combinatorial structures. We establish various results of different natures, in both graph and hypergraph theory, most of them using the absorbing method: 1. We prove an asymptotically best-possible bound on the strong chromatic number with respect to the maximum degree of the graph. This establishes a weak version of a conjecture of Aharoni, Berger and Ziv. 2. We determine asymptotic minimum codegree thresholds which ensure the existence of tilings with tight cycles (of a given size) in uniform hypergraphs. Moreover, we prove results on coverings with tight cycles. 3. We show that every 2-coloured complete graph on the integers contains a monochromatic infinite path whose vertex set is sufficiently "dense" in the natural numbers. This improves results of Galvin and Erdős and of DeBiasio and McKenney

Packings and tilings in dense graphs

In this thesis we present results on selected problems from extremal graph theory, and discuss both known and new methods used to solve them. In Chapter 1, we give an introductory overview of the regularity method, the flag algebra framework, and some probabilistic tools, which we use to prove our results in subsequent chapters. In Chapter 2 we prove a new result on the packing density of triangles in graphs with given edge density. In doing so, we settle a few conjectures of Gyori and Tuza on decompositions and coverings of graphs with cliques of bounded size. In Chapter 3 we show that a famous conjecture on Hamilton decompositions of bipartite tournaments due to Jackson holds approximately, providing the first intermediate result towards a full proof of the conjecture. In Chapter 4, we introduce a novel absorbing paradigm for graph tilings, which we apply in a few different settings to obtain new results. Using this method, we are able to extend a result on triangle-tilings in graphs with high minimum degree and sublinear independence number to clique-tilings of arbitrary size. We also strengthen an existing result on tilings in randomly perturbed graphs. Finally, in Chapter 5, we consider a problem on quasi-randomness in permutations. We obtain simple density conditions for a sequence of permutations to be quasirandom, and give a full characterisation of all conditions of the same type that force quasi-randomness in the same way

Extremal problems in disjoint cycles and graph saturation

In this thesis, we tackle two main themes: sufficient conditions for the existence of particular subgraphs in a graph, and variations on graph saturation. Determining whether a graph contains a certain subgraph is a computationally difficult problem; as such, sufficient conditions for the existence of a given subgraph are prized. In Chapter 2, we offer a significant refinement of the Corradi-Hajnal Theorem, which gives sufficient conditions for the existence of a given number of disjoint cycles in a graph. Further, our refined theorem leads to an answer for a question posed by G. Dirac in 1963 regarding the existence of disjoint cycles in graphs with a certain connectivity. This answer comprises Chapter 3. In Chapter 4 we prove a result about equitable coloring: that is, a proper coloring whose color classes all have the same size. Our equitable-coloring result confirms a partial case of a generalized version of the much-studied Chen-Lih-Wu conjecture on equitable coloring. In addition, the equitable-coloring result is equivalent to a statement about the existence of disjoint cycles, contributing to our refinement of the Corradi-Hajnal Theorem. In Chapters 5 and 6, we move to the topic of graph saturation, which is related to the Turan problem. One imagines a set of n vertices, to which edges are added one-by-one so that a forbidden subgraph never appears. At some point, no more edges can be added. The Turan problem asks the maximum number of edges in such a graph; the saturation number, on the other hand, asks the minimum number of edges. Two variations of this parameter are studied. In Chapter 5, we study the saturation of Ramsey-minimal families. Ramsey theory deals with partitioning the edges of graphs so that each partition avoids the particular forbidden subgraph assigned to it. Our motivation for studying these families is that they provide a convincing edge-colored (Ramsey) version of graph saturation. We develop a method, called iterated recoloring, for using results from graph saturation to understand this Ramsey version of saturation. As a proof of concept, we use iterated recoloring to determine the saturation number of the Ramsey-minimal families of matchings and describe the assiociated extremal graphs. An induced version of graph saturation was suggested by Martin and Smith. In order to offer a parameter that is defined for all forbidden graphs, Martin and Smith consider generalized graphs, called trigraphs. Of particular interest is the case when the induced-saturated trigraphs in question are equivalent to graphs. In Chapter 6, we show that a surprisingly large number of families fall into this case. Further, we define and investigate another parameter that is a version of induced saturation that is closer in spirit to the original version of graph saturation, but that is not defined for all forbidden subgraphs