Unavoidable subgraphs of colored graphs

A natural generalization of graph Ramsey theory is the study of unavoidable sub-graphs in large colored graphs. In this paper, we find a minimal family of unavoidable graphs in two-edge-colored graphs. Namely, for a positive even integer k, let S k be the family of two-edge-colored graphs on k vertices such that one of the colors forms either two disjoint K k / 2 's or simply one K k / 2 . Bollobás conjectured that for all k and ε > 0 , there exists an n ( k , ε ) such that if n ⩾ n ( k , ε ) then every two-edge-coloring of K n , in which the density of each color is at least ε , contains a member of this family. We solve this conjecture and present a series of results bounding n ( k , ε ) for different ranges of ε . In particular, if ε is sufficiently close to 1 2 , the gap between our upper and lower bounds for n ( k , ε ) is smaller than those for the classical Ramsey number R ( k , k ) .