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