Highly linked tournaments

A (possibly directed) graph is k-linked if for any two disjoint sets of vertices {x1,…,xk} and {y1,…,yk} there are vertex disjoint paths P1,…,Pk such that Pi goes from xi to yi. A theorem of Bollobás and Thomason says that every 22k-connected (undirected) graph is k-linked. It is desirable to obtain analogues for directed graphs as well. Although Thomassen showed that the Bollobás-Thomason Theorem does not hold for general directed graphs, he proved an analogue of the theorem for tournaments - there is a function f(k) such that every strongly f(k)-connected tournament is k-linked. The bound on f(k) was reduced to O(klogk) by Kühn, Lapinskas, Osthus, and Patel, who also conjectured that a linear bound should hold. We prove this conjecture, by showing that every strongly 452k-connected tournament is k-linked

Edge disjoint Hamiltonian cycles in highly connected tournaments

Thomassen conjectured that there is a function f(k) such that every strongly f(k)-connected tournament contains k edge-disjoint Hamiltonian cycles. This conjecture was recently proved by Kühn, Lapinskas, Osthus, and Patel who showed that f(k) ≤ O(k 2 (logk) 2 ) and conjectured that there is a constant C such that f(k) ≤ Ck 2 . We prove this conjecture. As a second application of our methods we answer a question of Thomassen about spanning linkages in highly connected tournaments

On the structure of dense graphs, and other extremal problems

Extremal combinatorics is an area of mathematics populated by problems that are easy to state, yet often difficult to resolve. The typical question in this field is the following: What is the maximum or minimum size of a collection of finite objects (e.g., graphs, finite families of sets) subject to some set of constraints? Despite its apparent simplicity, this question has led to a rather rich body of work. This dissertation consists of several new results in this field.The first two chapters concern structural results for dense graphs, thus justifying the first part of my title. In the first chapter, we prove a stability result for edge-maximal graphs without complete subgraphs of fixed size, answering questions of Tyomkyn and Uzzell. The contents of this chapter are based on joint work with Kamil Popielarz and Julian Sahasrabudhe.The second chapter is about the interplay between minimum degree and chromatic number in graphs which forbid a specific set of `small\u27 graphs as subgraphs. We determine the structure of dense graphs which forbid triangles and cycles of length five. A particular consequence of our work is that such graphs are 3-colorable. This answers questions of Messuti and Schacht, and Oberkampf and Schacht. This chapter is based on joint work with Shoham Letzter.Chapter 3 departs from undirected graphs and enters the domain of directed graphs. Specifically, we address the connection between connectivity and linkedness in tournaments with large minimum out-degree. Making progress on a conjecture of Pokrovskiy, we show that, for any positive integer $k$, any $4k$-connected tournament with large enough minimum out-degree is $k$-linked. This chapter is based on joint work with Ant{\\u27o}nio Gir{\~a}o.ArrayThe final chapter leaves the world of graphs entirely and examines a problem in finite set systems.More precisely, we examine an extremal problem on a family of finite sets involving constraints on the possible intersectionsizes these sets may have. Such problems have a long history in extremal combinatorics. In this chapter, we are interested in the maximum number of disjoint pairs a family of sets can have under various restrictions on intersection sizes. We obtain several new results in this direction. The contents of this chapter are based on joint work with Ant{\\u27o}nio Gir{\~a}o