16 research outputs found
Upper density problems in infinite Ramsey theory
We consider the following question in infinite Ramsey theory, introduced by ErdĆs 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 ErdĆs 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 ErdĆs und Galvin [EG93] in einem Spezialfall sowie von DeBiasio und McKenney [DM19] in einem allgemeineren Kontext formuliert wurde: Es sei H ein abzĂ€hlbar unendlicher Graph. Welches ist der gröĂtmögliche Wert λ, sodass wir, wenn die Kanten des vollstĂ€ndigen Graphen mit Knotenmenge N jeweils entweder rot oder blau gefĂ€rbt 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 ErdĆs 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 unabhĂ€ngigen Mengen. Wir ermitteln die exakten Werte der Ramsey-Dichte fĂŒr eine Vielzahl von Graphen, darunter alle lokal endlichen WĂ€lder, bipartite Faktoren, Kr-Faktoren sowie Ck-Faktoren fĂŒr ungerade k. Ferner bestimmen wir den Wert der Ramsey-Dichte aller lokal endlichen Graphen bis auf einen Faktor 2.
DarĂŒber hinaus untersuchen wir eine Variante des oben beschriebenen Problems fĂŒr ListenfĂ€rbungen. Wir zeigen, dass es möglich ist, jeder Kante des vollstĂ€ndigen Graphen mit Knotenmenge N eine Liste der GröĂe Zwei zuzuweisen, sodass in jeder zugehörigen ListenfĂ€rbung monochromatische Pfade mit beliebig nah an 1 liegender Dichte existieren
Planar graph with twin-width seven
We construct a planar graph with twin-width equal to seven
A Note on the Minimum Number of Edges in Hypergraphs with Property O
An oriented -uniform hypergraph is said to have Property O if for every
linear order of the vertex set, there is some edge oriented consistently with
the linear order. Recently Duffus, Kay and R\"{o}dl investigated the minimum
number of edges in a -uniform hypergaph with Property O. They proved
that , where the upper bound holds for
sufficiently large. In this short note we improve their upper bound by a factor
of , showing that for every . We also
show that their lower bound is not tight. Furthermore, Duffus, Kay and R\"{o}dl
also studied the minimum number of vertices in a -uniform hypergaph
with Property O. For they showed , and asked for
the precise value of . Here we show .Comment: 6 pages, 1 figur
Removal lemmas in sparse graphs
In this work we explain and prove the graph removal lemma, both in its dense and sparse cases, and show how these can be applied to finite groups to obtain arithmetic removal lemmas. We show how the concept of regularity plays a crucial role in the proof of the removal lemma. We explain the motivation behind the sparse case, and the importance of pseudorandom graphs in sparse versions of the removal lemma. Finally, we show how the removal lemma, both in its graph and arithmetic versions, can be used to prove Roth's theorem, that is, the existence of 3-term arithmetic progressions in any dense subset of the natural numbers
The dimension of the region of feasible tournament profiles
Erd\H os, Lov\'asz and Spencer showed in the late 1970s that the dimension of
the region of -vertex graph profiles, i.e., the region of feasible densities
of -vertex graphs in large graphs, is equal to the number of non-trivial
connected graphs with at most vertices. We determine the dimension of the
region of -vertex tournament profiles. Our result, which explores an
interesting connection to Lyndon words, yields that the dimension is much
larger than just the number of strongly connected tournaments, which would be
the answer expected as the analogy to the setting of graphs
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+ â 8)/17 â 0.87226 and further show that this is best possible. This settles a problem of Erdos and Galvi