Unavoidable subgraphs of colored graphs

A natural generalization of graph Ramsey theory is the study of unavoidable subgraphs in large colored graphs. In this paper, we find a minimal family of unavoidable graphs in two-edgecolored graphs. Namely, for a positive even integer k, let Sk be the family of two-edge-colored graphs on k vertices such that one of the colors forms either two disjoint Kk/2s or simply one Kk/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-colorings of Kn, in which the density of each color is at least ε, contain 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).