Small Ramsey Numbers

We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values

THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

On the Ramsey numbers for stars versus complete graphs

For graphs G1, . . . , Gs, the multicolor Ramsey number R(G1, . . . , Gs) is the smallest integer r such that if we give any edge col-oring of the complete graph on r vertices with s colors then there exists a monochromatic copy of Gi colored with color i, for some 1 ≤ i ≤ s. In this work the multicolor Ramsey number R(Kp1 , . . . , Kpm , K1,q1 , . . . , K1,qn ) is determined for any set of com-plete graphs and stars in terms of R(Kp1 , . . . , Kpm )Ministerio de Educación y Ciencia MTM2008-06620-C03-02Junta de Andalucía P06-FQM-0164

Induced Ramsey number for a star versus a fixed graph

For graphs G and H, let the induced Ramsey number IR(H,G) be the smallest number of vertices in a graph F such that any coloring of the edges of F in red and blue, there is either a red induced copy of H or a blue induced copy of G. In this note we consider the case when G=Sn is a star on n edges, for large n, and H is a fixed graph. We prove that (r-1)n < IR(H, Sn) < (r-1)(r-1)n + cn, for any c>0, sufficiently large n, and r denoting the chromatic number of H. The lower bound is asymptotically tight for any fixed bipartite H. The upper bound is attained up to a constant factor, for example by a clique H