A uniform hypergraph H is called k-Ramsey for a hypergraph F , if no matter how one colors the edges of H with k colors, there is always a monochromatic copy of F . We say that H is minimal k-Ramsey for F , if H is k-Ramsey for F but every proper subhypergraph of H is not. Burr, Erdős and Lovasz [S. A. Burr, P. Erdős, and L. Lovász, On graphs of Ramsey type, Ars Combinatoria 1 (1976), no. 1, 16...

An ordered graph is a graph equipped with a linear ordering of its vertex set. A pair of ordered graphs is Ramsey finite if it has only finitely many minimal ordered Ramsey graphs and Ramsey infinite otherwise. Here an ordered graph F is an ordered Ramsey graph of a pair (H,H ′) of ordered graphs if for any coloring of the edges of F in colors red and blue there is either a copy of H with all e...

We want to thank the authors for responding to our paper and for apprising readers as to the effects of the new information technologies in Asian educational settings. We think Miyake has characterized our argument very accurately and succinctly. As she points out the equity issues we raise are exacerbated in Asia because the resources available through new technologies are mostly in English. B...

There are two equivalent problem statements for the Ramsey number n = R(k; l). n is the minimum number of vertices in the graph such that it contains a complete graph of k vertices, or an independent set of l vertices. n is the minimum number of vertices such that if all the edges of the complete graph on n vertices, denoted by K n is colored with two colors, fRed, Blueg, then there exists a Re...

We present a refinement of Ramsey numbers by considering graphs with a partial ordering on their vertices. This is a natural extension of the ordered Ramsey numbers. We formalize situations in which we can use arbitrary families of partially-ordered sets to form host graphs for Ramsey problems. We explore connections to well studied Turán-type problems in partially-ordered sets, particularly th...

We introduce a new construct that can be used to parametrize a topological Ramsey space by the collection of infinite subsets of natural numbers. We show that these parametrized spaces are topological Ramsey spaces. Then we prove a canonical Ramsey theorem for some of the parametrized spaces arising from the construction and conclude with some open questions concerning applications of these can...

For a k-uniform hypergraph G with vertex set {1, . . . , n}, the ordered Ramsey number ORt(G) is the least integer N such that every t-coloring of the edges of the complete k-uniform graph on vertex set {1, . . . , N} contains a monochromatic copy of G whose vertices follow the prescribed order. Due to this added order restriction, the ordered Ramsey numbers can be much larger than the usual gr...