Given a topological Ramsey space (R,≤, r), we extend the notion of semiselective coideal to sets H ⊆ R and study conditions for H that will enable us to make the structure (R,H,≤, r) a Ramsey space (not necessarily topological) and also study forcing notions related to H which will satisfy abstract versions of interesting properties of the corresponding forcing notions in the realm of Ellentuck...

We use nonstandard methods, based on iterated hyperextensions, to develop applications to Ramsey theory of the theory of monads of ultrafilters. This is performed by studying in detail arbitrary tensor products of ultrafilters, as well as by characterizing their combinatorial properties by means of their monads. This extends to arbitrary sets methods previously used to study partition regular D...

Abstract The size-Ramsey number of a graph F is the smallest edges in G with Ramsey property for , that is, any 2-colouring contains monochromatic copy . We prove grid on n × vertices bounded from above by 3+o(1)

we canvas a number of past uses of Ramsey sentences which have yielded disappointing results, and then consider three very interesting recent attempts to deploy them for a Ramseyan dialetheist theory of truth, a modal account of laws and theories, and a criterion for the existence of factual properties. we think that once attention is given to the specific kinds of theories that Ramsey had in m...

Associated to each ultrafilter U on ω and each map p : ω → ω is a Dedekind cut in the ultrapower ω/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 spaceR1. Associated to the spaceR1 is a notion of Ramsey ultrafilter ...

For given graphs G and H, the Ramsey number R(G, H) is the least natural number n such that for every graph F of order n the following condition holds: either F contains G or the complement of F contains H. In this paper, we improve the Surahmat and Tomescu’s result [9] on the Ramsey number of paths versus Jahangirs. We also determine the Ramsey number R(∪G, H), where G is a path and H is a Jah...

The Ramsey multiplicity R(G) of a graph G is the minimum number of monochromatic copies of G in any two-colouring of the edges of Kr(G), where r(G) denotes the Ramsey number of G. Here we prove that odd cycles have super-exponentially large Ramsey multiplicity: If Cn is an odd cycle of length n, then logR(Cn) = Θ(n logn).

We say that a graph G has the Ramsey property w.r.t. some graph F and some integer r ≥ 2, or G is (F, r)-Ramsey for short, if any r-coloring of the edges of G contains a monochromatic copy of F . Rödl and Ruciński asked how globally sparse (F, r)-Ramsey graphs G can possibly be, where the density of G is measured by the subgraph H ⊆ G with the highest average degree. So far, this so-called Rams...