The Ramsey number of dense graphs
Abstract
The Ramsey number r(H) of a graph H 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 , proving that r(H) ≤ 2c (2/) t. We also investigate some related problems, such as the Ramsey number of graphs with t vertices and maximum degree t and the Ramsey number of random graphs in G(t, ), that is, graphs on t vertices where each edge has been chosen independently with probability .
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