The codegree threshold of K4−K_4^-

The codegree threshold $\mathrm{ex}_2(n, F)$ of a $3$-graph $F$ is the minimum $d=d(n)$ such that every $3$-graph on $n$ vertices in which every pair of vertices is contained in at least $d+1$ edges contains a copy of $F$ as a subgraph. We study $\mathrm{ex}_2(n, F)$ when $F=K_4^-$, the $3$-graph on $4$ vertices with $3$ edges. Using flag algebra techniques, we prove that if $n$ is sufficiently large then $\mathrm{ex}_2(n, K_4^-)\leq (n+1)/4$. This settles in the affirmative a conjecture of Nagle from 1999. In addition, we obtain a stability result: for every near-extremal configuration $G$, there is a quasirandom tournament $T$ on the same vertex set such that $G$ is close in the edit distance to the $3$-graph $C(T)$ whose edges are the cyclically oriented triangles from $T$. For infinitely many values of $n$, we are further able to determine $\mathrm{ex}_2(n, K_4^-)$ exactly and to show that tournament-based constructions $C(T)$ are extremal for those values of $n$.Comment: 31 pages, 7 figures. Ancillary files to the submission contain the information needed to verify the flag algebra computation in Lemma 2.8. Expands on the 2017 conference paper of the same name by the same authors (Electronic Notes in Discrete Mathematics, Volume 61, pages 407-413

Colouring versus density in integers and Hales-Jewett cubes

We construct for every integer $k\geq 3$ and every real $\mu\in(0, \frac{k-1}{k})$ a set of integers $X=X(k, \mu)$ which, when coloured with finitely many colours, contains a monochromatic $k$-term arithmetic progression, whilst every finite $Y\subseteq X$ has a subset $Z\subseteq Y$ of size $|Z|\geq \mu |Y|$ that is free of arithmetic progressions of length $k$. This answers a question of Erd\H{o}s, Ne\v{s}et\v{r}il, and the second author. Moreover, we obtain an analogous multidimensional statement and a Hales-Jewett version of this result.Comment: 5 figure

Tilings in randomly perturbed graphs: Bridging the gap between Hajnal‐Szemerédi and Johansson‐Kahn‐Vu

A perfect Kr-tiling in a graph G is a collection of vertex-disjoint copies of Kr that together cover all the vertices in G. In this paper we consider perfect Kr-tilings in the setting of randomly perturbed graphs; a model introduced by Bohman, Frieze, and Martin [7] where one starts with a dense graph and then adds m random edges to it. Specifically, given any fixed 0 < < 1 − 1∕r we determine how many random edges one must add to an n-vertex graph G of minimum degree (G) ≥ n to ensure that, asymptotically almost surely, the resulting graph contains a perfect Kr-tiling. As one increases we demonstrate that the number of random edges required “jumps” at regular intervals, and within these intervals our result is best-possible. This work therefore closes the gap between the seminal work of Johansson, Kahn and Vu [25] (which resolves the purely random case, that is, = 0) and that of Hajnal and Szemerédi [18] (which demonstrates that for ≥ 1 − 1∕r the initial graph already houses the desired perfect Kr-tiling)

Problems in Extremal Graph Theory

This dissertation consists of six chapters concerning a variety of topics in extremal graph theory.Chapter 1 is dedicated to the results in the papers with Antnio Giro, Gbor Mszros, and Richard Snyder. We say that a graph is path-pairable if for any pairing of its vertices there exist edge disjoint paths joining the vertices in eachpair. We study the extremal behavior of maximum degree and diameter in some classes of path-pairable graphs. In particular we show that a path-pairable planar graph must have a vertex of linear degree.In Chapter 2 we present a joint work with Antnio Giro and Teeradej Kittipassorn. Given graphs G and H, we say that a graph F is H-saturated in G if F is H-free subgraph of G, but addition of any edge from E(G) to F creates a copy of H. Here we deal with the case when G is a complete k-partite graph with n vertices in each class, and H is a complete graph on r vertices. We prove bounds, which are tight for infinitely many values of k and r, on the minimal number of edges in a H-saturated graph in G, for this choice of G and H, answering questions of Ferrara, Jacobson, Pfender, and Wenger; and generalizing a result of Roberts.Chapter 3 is about a joint paper with Antnio Giro and Teeradej Kittipassorn. A coloring of the vertices of a digraph D is called majority coloring if no vertex of D receives the same color as more than half of its outneighbours. This was introduced by van der Zypen in 2016. Recently, Kreutzer, Oum, Seymour, van der Zypen, and Wood posed a number of problems related to this notion: here we solve several of them.In Chapter 4 we present a joint work with Antnio Giro. We show that given any integer k there exist functions g1(k), g2(k) such that the following holds. For any graph G with maximum degree one can remove fewer than g1(k) ^{1/2} vertices from G so that the remaining graph H has k vertices of the same degree at least (H) g2(k). It is an approximate version of conjecture of Caro and Yuster; and Caro, Lauri, and Zarb, who conjectured that g2(k) = 0.Chapter 5 concerns results obtained together with Kazuhiro Nomoto, Julian Sahasrabudhe, and Richard Snyder. We study a graph parameter, the graph burning number, which is supposed to measure the speed of the spread of contagion in a graph. We prove tight bounds on the graph burning number of some classes of graphs and make a progress towards a conjecture of Bonato, Janssen, and Roshanbin about the upper bound of graph burning number of connected graphs.In Chapter 6 we present a joint work with Teeradej Kittipassorn. We study the set of possible numbers of triangles a graph on a given number of vertices can have. Among other results, we determine the asymptotic behavior of the smallest positive integer m such that there is no graph on n vertices with exactly m copies of a triangle. We also prove similar results when we replace triangle by any fixed connected graph