It is shown that the Ramsey number of any graph with n vertices in which no two vertices of degree at least 3 are adjacent is at most 12n. In particular, the above estimate holds for the Ramsey number of any n-vertex subdivision of an arbitrary graph, provided each edge of the original graph is subdivided at least once. This settles a problem of Burr and Erdös.

Given any graph $H$, a $G$ is said to be $q$-Ramsey for $H$ if every coloring of the edges with $q$ colors yields monochromatic subgraph isomorphic $H$. Such minimal additionally no proper $G'$ In 1976, Burr, Erdös, and Lovász initiated study parameter $s_q(H)$, defined as smallest minimum degree among all graphs this paper, we consider problem determining how many vertices $s_q(H)$ can contain...

Recently we determined the Ramsey Number r(C7, C7, C7) = 25. Let G = (V (G), E(G)) be an undirected finite graph without any loops or multiple edges, where V (G) denotes its vertex set and E(G) its edge set. In the following we will often consider the complete graph Kp on p vertices and the cycle Cp on p vertices. A k−coloring (F1, F2, . . . , Fk) of a graph G is a coloring of the edges of G wi...

appeared in the proceedings of ICDT’10. [44] M. Bodirsky and J. Nešetřil. Constraint satisfaction with countable homogeneous templates. In Proceedings of CSL, pages 44–57, Vienna, 2003. [45] M. Bodirsky and J. Nešetřil. Constraint satisfaction with countable homogeneous templates. Journal of Logic and Computation, 16(3):359–373, 2006. [46] M. Bodirsky and D. Piguet. Finite trees are Ramsey with...

Let G and H be simple graphs. The Ramsey number r(G,H) for a pair of graphs G and H is the smallest number r such that any red-blue coloring of the edges of Kr contains a red subgraph G or a blue subgraph H . The size Ramsey number r̂(G,H) for a pair of graphs G and H is the smallest number r̂ such that there exists a graph F with size r̂ satisfying the property that any red-blue coloring of the e...

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...

Given a graph H and an integer r ≥ 2, let G → (H, r) denote the Ramsey property of a graph G, that is, every r-coloring of the edges of G results in a monochromatic copy of H. Further, let m(G) = maxF⊆G |E(F )|/|V (F )| and define the Ramsey density minf (H, r) as the infimum of m(G) over all graphs G such that G → (H, r). In the first part of this paper we show that when H is a complete graph ...

In this paper, we compare the offline versions of three Ramsey-type oneplayer games that have been studied in an online setting in previous work: the online Ramsey game, the balanced online Ramsey game, and the Achlioptas game. The goal in all games is to color the edges of the random graph Gn,m according to certain rules without creating a monochromatic copy of some fixed forbidden graph H. Wh...

Let k be a fixed positive integer and let H be a graph with at least k + 1 edges. A local (H, k)-coloring of a graph G is a coloring of the edges of G such that edges of no subgraph of G isomorphic to a subgraph of H are colored with more than k colors. In the paper we investigate properties of local (H, k)-colorings. We prove the Ramsey property for such colorings, establish conditions for the...