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...

Given a graph H, a graph G is called a Ramsey graph of H if there is a monochromatic copy of H in every coloring of the edges of G with two colors. Two graphs G, H are called Ramsey equivalent if they have the same set of Ramsey graphs. Fox et al. [J. Combin. Theory Ser. B 109 (2014), 120–133] asked whether there are two nonisomorphic connected graphs that are Ramsey equivalent. They proved tha...

In this paper we introduce a new topological Ramsey space P whose elements are infinite ordered polyhedra. The corresponding familiy AP of finite approximations can be viewed as a class of finite structures. It turns out that the closure of AP under isomorphisms is the class KP of finite ordered polyhedra. Following [8], we show that KP is a Ramsey class. Then, we prove a universal property for...

‎in this paper‎, ‎we define the common minimal dominating signed‎ ‎graph of a given signed graph and offer a structural‎ ‎characterization of common minimal dominating signed graphs‎. ‎in‎ ‎the sequel‎, ‎we also obtained switching equivalence‎ ‎characterizations‎: ‎$overline{s} sim cmd(s)$ and $cmd(s) sim‎ ‎n(s)$‎, ‎where $overline{s}$‎, ‎$cmd(s)$ and $n(s)$ are complementary‎ ‎signed gra...

We show that if G is a Ramsey size-linear graph and x,y 2 V(G ) then if we add a sufficiently long path between x and y we obtain a new Ramsey size-linear graph. As a consequence we show that if G is any graph such that every cycle in G contains at least four consecutive vertices of degree 2 then G is Ramsey size-linear. ß 2002 John Wiley & Sons, Inc. J Graph Theory 39: 1–5, 2002

The size-Ramsey number of a graph G is the minimum number of edges in a graph H such that every 2-edge-coloring of H yields a monochromatic copy of G. Size-Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs. We initiate the study of size-Ramsey numbers for k-uniform hypergraphs. Analogous to the graph case, we cons...

A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number, and Ramsey saturated otherwise. We present some result concerning both Ramsey saturated and unsaturated graph. In particular, we show that a cycle Cn and a Jahangir Jm Ramsey unsaturated or saturated graphs of R(Cn,Wm) and R(Pn, Jm), respectively. We also suggest an open problems.

For graphs G,F and H we write G → (F,H) to mean that if the edges of G are coloured with two colours, say red and blue, then the red subgraph contains a copy of F or the blue subgraph contains a copy of H. The graph G is (F,H)-minimal (Ramsey-minimal) if G → (F,H) but G′ 6→ (F,H) for any proper subgraph G′ ⊆ G. The class of all (F,H)-minimal graphs shall be denoted by R(F,H). In this paper we w...

An ordered graph G< is a graph G with vertices ordered by the linear ordering <. The ordered Ramsey number R(G<, c) is the minimum number N such that every ordered complete graph with c-colored edges and at least N vertices contains a monochromatic copy of G<. For unordered graphs it is known that Ramsey numbers of graphs with degrees bounded by a constant are linear with respect to the number ...

Given a labeled graph H with vertex set {1, 2, . . . , n}, the ordered Ramsey number r<(H) is the minimum N such that every two-coloring of the edges of the complete graph on {1, 2, . . . , N} contains a copy of H with vertices appearing in the same order as in H. The ordered Ramsey number of a labeled graph H is at least the Ramsey number r(H) and the two coincide for complete graphs. However,...