Journal ArticleDOI
Cube Ramsey numbers are polynomial
TLDR
It is shown that for any positive constant c and bipartite graph G=(U, V; E) of order n where the maximum degree of vertices in U is at most, the Ramsey number of the n-cube Qn satisfies which improves the bound 2cn log n due to Graham, Rodl, and Rucinski.Abstract:
Let R(G) be the least integer p such that for all bicolorings of the edges of complete graph Kp, at least one of the monochromatic subgraphs contains a copy of G. We show that for any positive constant c and bipartite graph G=(U, V; E) of order n where the maximum degree of vertices in U is at most In particular, this shows that the Ramsey number of the n-cube Qn satisfies which improves the bound 2cn log n due to Graham, Rodl, and Rucinski. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19: 99–101, 2001read more
Citations
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Journal ArticleDOI
Small Ramsey Numbers
TL;DR: This work presents data which, to the best of its 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.
Book ChapterDOI
Embedding large subgraphs into dense graphs
Daniela Kühn,Deryk Osthus +1 more
TL;DR: In this article, the authors give an overview of recent progress in F-packings, with the main emphasis on F-packing, Hamiltonicity problems and tree embeddings, and describe some of the methods involved.
Journal ArticleDOI
Density theorems for bipartite graphs and related Ramsey-type results
Jacob Fox,Benny Sudakov +1 more
TL;DR: In this article, the authors present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density using probabilistic arguments with some combinatorial ideas.
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Dependent random choice
Jacob Fox,Benny Sudakov +1 more
TL;DR: In this paper, a simple and yet surprisingly powerful probabilistic technique which shows how to find in a dense graph a large subset of vertices in which all (or almost all) small subsets have many common neighbors is described.
Book ChapterDOI
Recent developments in graph Ramsey theory.
TL;DR: There has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics.
References
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Journal ArticleDOI
On graphs with linear Ramsey numbers
TL;DR: For a fixed graph H, the Ramsey number r(H) is defined to be the least integer n such that in any 2-coloring of the edges of the complete graph KN, some monochromatic copy of H is always formed.
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