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Keywords:

  • size-Ramsey number;
  • trees;
  • expander graphs

Abstract

Given a graph G, the size-Ramsey number equation image is the minimum number m for which there exists a graph F on m edges such that any two-coloring of the edges of F admits a monochromatic copy of G.

In 1983, J. Beck introduced an invariant β(·) for trees and showed that equation image. Moreover he conjectured that equation image. We settle this conjecture by providing a family of graphs and an embedding scheme for trees. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011