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{lunate} > 0, there exists an n (k, ε{lunate}) such that if n ≥ n (k, ε{lunate}) then every two-edge-coloring of K_n, in which the density of each color is at least ε{lunate}, contains a member of this family. We solve this conjecture and present a series of results bounding n (k, ε{lunate}) for different ranges of ε{lunate}. In particular, if ε{lunate} is sufficiently close to frac(1, 2), the gap between our upper and lower bounds for n (k, ε{lunate}) is smaller than those for the classical Ramsey number R (k, k).