Density of Monochromatic Infinite Subgraphs

Abstract

For any countably infinite graph G, Ramsey's theorem guarantees an infinite monochromatic copy of G in any r-coloring of the edges of the countably infinite complete graph K N . Taking this a step further, it is natural to wonder how "large" of a monochromatic copy of G we can find with respect to some measure -for instance, the density (or upper density) of the vertex set of G in the positive integers. Unlike finite Ramsey theory, where this question has been studied extensively, the analogous problem for infinite graphs has been mostly overlooked.In one of the few results in the area, Erdős and Galvin proved that in every 2-coloring of K N , there exists a monochromatic path whose vertex set has upper density at least 2/3, but it is not possible to do better than 8/9. They also showed that for some sequence n → 0, there exists a monochromatic path P such that for infinitely many n, the set {1, 2, ..., n} contains the first ( 1 3+ √ 3 − n )n vertices of P , but it is not possible to do better than 2n/3. We improve both results, in the former case achieving an upper density at least 3/4 and in the latter case obtaining a tight bound of 2/3. We also consider related problems for directed paths, trees (connected subgraphs), and a more general result which includes locally finite graphs for instance.

Partitioning edge-coloured complete symmetric digraphs into monochromatic complete subgraphs

“…By the length of a path, we mean the number of edges in the path. DeBiasio and McKenney [3] recently proved the following result.…”

Upper density of monochromatic infinite paths

“…In this section we use Lemma 4.2 to prove Theorem 1.2. Our exposition follows that of Theorem 1.6 in [2].…”

“…For ≥ 4, let {W j } j∈[ ] be a partition of [n] such that each W j consists of at most n/ subsequent vertices. The partition {W j } j∈ [2 ] , with parts of the form W i ∩ R and W i ∩ B, refines both {W j } j∈[ ] and {R, B}. Suppose that V 0 ∪ · · · ∪ V m is a partition obtained from Lemma 4.6 applied to G and {W j } j∈ [2 ] with parameters ε, m 0 , 2 and d. We define the (ε, d)-reduced graph G to be the graph with vertex set…”

“…Moreover, they constructed a 2-colouring of K N in which every monochromatic path P has upper density at most 8/9. DeBiasio and McKenney [2] recently improved the lower bound to 3/4 and conjectured the correct value to be 8/9. Progress towards this conjecture was made by Lo, Sanhueza-Matamala and Wang [8], who raised the lower bound to ( Now that we have solved the problem for two colours, it would be very interesting to make any improvement on Rado's lower bound of 1/r for r ≥ 3 colours (see [2,Corollary 3.5] for the best known upper bound).…”

Large monochromatic components in 3‐edge‐colored Steiner triple systems

“…We will prove a lower bound on for all 3‐uniform hypergraphs with (ie, every pair of vertices is contained in at least one edge). We use the following lemma from [8] which we reproduce here for completeness (for technical reasons we allow for edges to receive multiple colors, so we have modified the statement of the lemma accordingly).…”