Upper density of monochromatic infinite paths

We prove that in every 2-colouring of the edges of KN 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 Erdos and Galvi

Density of monochromatic infinite paths

For any subset $A \subseteq \mathbb{N}$, we define its upper density to be $\limsup_{ n \rightarrow \infty } |A \cap \{ 1, \dotsc, n \}| / n$. We prove that every $2$-edge-colouring of the complete graph on $\mathbb{N}$ contains a monochromatic infinite path, whose vertex set has upper density at least $(9 + \sqrt{17})/16 \approx 0.82019$. This improves on results of Erd\H{o}s and Galvin, and of DeBiasio and McKenney.Comment: Accepted for publication in The Electronic Journal of Combinatoric

Upper density problems in infinite Ramsey theory

We consider the following question in infinite Ramsey theory, introduced by Erdős and Galvin [EG93] in a particular case and by DeBiasio and McKenney [DM19] in a more general setting. Let H be a countably infinite graph. If the edges of the complete graph on the natural numbers are colored red or blue, what is the maximum value of λ such that we are guaranteed to find a monochromatic copy of H whose vertex set has upper density at least λ? We call this value the Ramsey density of H. The problem of determining the Ramsey density of the infinite path was first studied by Erdős and Galvin, and was recently solved by Corsten, DeBiasio, Lang and the author [CDLL19]. In this thesis we study the problem of determining the Ramsey density of arbitrary graphs H. On an intuitive level, we show that three properties of a graph H have an effect on the Ramsey density: the chromatic number, the number of components, and the expansion of its independent sets. We deduce the exact value of the Ramsey density for a wide variety of graphs, including all locally finite forests, bipartite factors, clique factors and odd cycle factors. We also determine the value of the Ramsey density of all locally finite graphs, up to a factor of 2. We also study a list coloring variant of the same problem. We show that there exists a way of assigning a list of size two to every edge in the complete graph on N such that, in every list coloring, there are monochromatic paths with density arbitrarily close to 1.Wir betrachten die folgende Fragestellung aus der Ramsey-Theorie, welche von Erdős und Galvin [EG93] in einem Spezialfall sowie von DeBiasio und McKenney [DM19] in einem allgemeineren Kontext formuliert wurde: Es sei H ein abzählbar unendlicher Graph. Welches ist der größtmögliche Wert λ, sodass wir, wenn die Kanten des vollständigen Graphen mit Knotenmenge N jeweils entweder rot oder blau gefärbt sind, stets eine einfarbige Kopie von H, dessen Knotenmenge eine obere asymptotische Dichte von mindestens λ besitzt, finden können? Wir nennen diesen Wert die Ramsey-Dichte von H. Das Problem, die Ramsey-Dichte des unendlichen Pfades zu bestimmen wurde erstmals von Erdős und Galvin untersucht und wurde vor kurzem von Corsten, DeBiasio, Lang und dem Autor [CDLL19] gelöst. Gegenstand der vorliegenden Dissertation ist die Bestimmung der Ramsey-Dichten von Graphen. Auf einer intuitiven Ebene zeigen wir, dass drei Parameter eines Graphen die Ramsey-Dichte beeinflussen: die chromatische Zahl, die Anzahl der Zusammenhangskomponenten sowie die Expansion seiner unabhängigen Mengen. Wir ermitteln die exakten Werte der Ramsey-Dichte für eine Vielzahl von Graphen, darunter alle lokal endlichen Wälder, bipartite Faktoren, Kr-Faktoren sowie Ck-Faktoren für ungerade k. Ferner bestimmen wir den Wert der Ramsey-Dichte aller lokal endlichen Graphen bis auf einen Faktor 2. Darüber hinaus untersuchen wir eine Variante des oben beschriebenen Problems für Listenfärbungen. Wir zeigen, dass es möglich ist, jeder Kante des vollständigen Graphen mit Knotenmenge N eine Liste der Größe Zwei zuzuweisen, sodass in jeder zugehörigen Listenfärbung monochromatische Pfade mit beliebig nah an 1 liegender Dichte existieren

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