The $r$-size-Ramsey number $\hat{R}_r(H)$ of a graph $H$ is the smallest edges $G$ can have, such that for every edge-coloring with $r$ colors there exists monochromatic copy in $G$. notion size-Ramsey numbers has been introduced by Erdős, Faudree, Rousseau and Schelp 1978, attracted lot attention ever since. For $H$, we denote $H^q$ obtained from subdividing its $q{-}1$ vertices each. In recen...

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...

The irredundant Ramsey number s = s(m,n) [upper domination Ramsey number u = u(m,n), respectively] is the smallest natural number s [u, respectively] such that in any red-blue edge colouring (R,B) of the complete graph of order s [u, respectively], it holds that IR(B) ≥ m or IR(R) ≥ n [Γ(B) ≥ m or Γ(R) ≥ n, respectively], where Γ and IR denote respectively the upper domination number and the ir...

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...

The Ramsey number r(H) of a graphH is the smallest number n such that, in any two-colouring of the edges of Kn, there is a monochromatic copy of H . We study the Ramsey number of graphs H with t vertices and density ρ, proving that r(H) ≤ 2 √ ρ . We also investigate some related problems, such as the Ramsey number of graphs with t vertices and maximum degree ρt and the Ramsey number of random g...

In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniqu...

We show that several known Ramsey number inequalities can be extended to the setting of r-uniform hypergraphs. In particular, we extend Burr’s results on tree-star Ramsey numbers, providing exact evaluations for certain hypergraph Ramsey numbers. Then we turn our attention to proving a general multicolor hypergraph Ramsey number inequality from which generalizations of results due to Chvátal an...