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...

The Ramsey number R(m,n) is the smallest integer p such that any blue-red colouring of the edges of the complete graph Kp forces the appearance of a blue Km or a red Kn. Bipartite Ramsey problems deal with the same questions but the graph explored is the complete bipartite graph instead of the complete graph. We investigate the appearance of simpler monochromatic graphs such as stripes, stars a...

A graph is Ramsey unsaturated if there exists a proper supergraph of the same order with the same Ramsey number, and Ramsey saturated otherwise. This has been studied by Balister, Lehel and Schelp [J. Graph Theory 51 (2006), 22–32]. In this paper, we show that some circulant graphs, trees with diameter 3, and Kt,n ∪ mK1 for infinitely many t, n and m, are Ramsey unsaturated.

We say that a graph with n vertices is c-Ramsey if it does not contain either a clique or an independent set of size c logn. We define a CNF formula which expresses this property for a graph G. We show a superpolynomial lower bound on the length of resolution proofs that G is c-Ramsey, for every graph G. Our proof makes use of the fact that every c-Ramsey graph must contain a large subgraph wit...

in this paper, we define the common minimal common neighborhooddominating signed graph (or common minimal $cn$-dominating signedgraph) of a given signed graph and offer a structuralcharacterization of common minimal $cn$-dominating signed graphs.in the sequel, we also obtained switching equivalencecharacterization: $overline{sigma} sim cmcn(sigma)$, where$overline{sigma}$ and $cmcn(sigma)$ are ...