We reveal a connection between the incompressibility method and the Lovász local lemma in the context of Ramsey theory. We obtain bounds by repeatedly encoding objects of interest and thereby compressing strings. The method is demonstrated on the example of van der Waerden numbers. It applies to lower bounds of Ramsey numbers, large transitive subtournaments and other Ramsey phenomena as well.

Associated to each ultrafilter U on ω and each map p : ω → ω is a Dedekind cut in the ultrapower ωω/p(U). Blass has characterized, under CH, the cuts obtainable when U is taken to be either a p-point ultrafilter, a weakly-Ramsey ultrafilter or a Ramsey ultrafilter. Dobrinen and Todorcevic have introduced the topological Ramsey space R1. Associated to the space R1 is a notion of Ramsey ultrafilt...

We produce a first order proof of a famous combinatorial result, Ramsey Theorem for pairs and in two colors. Our goal is to find the minimal classical principle that implies the “miniature” version of Ramsey we may express in Heyting Arithmetic HA. We are going to prove that Ramsey Theorem for pairs with recursive assignments of two colors is equivalent in HA to the sub-classical principle Σ3-L...

In this paper a concept Q-Ramsey Class of graphs is introduced, where Q is a class of bipartite graphs. It is a generalization of wellknown concept of Ramsey Class of graphs. Some Q-Ramsey Classes of graphs are presented (Theorem 1 and 2). We proved that T 2, the class of all outerplanar graphs, is not D1-Ramsey Class (Theorem 3). This results leads us to the concept of acyclic reducible bounds...

We call a graph H Ramsey-unsaturated if there is an edge in the complement of H such that the Ramsey number r(H) of H does not change upon adding it to H. This notion was introduced by Balister, Lehel and Schelp in [2], where it is shown that cycles (except for C4) are Ramsey-unsaturated, and conjectured that, moreover, one may add any chord without changing the Ramsey number of the cycle Cn, u...