Unavoidable Subgraphs of Colored Graphs

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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 Sk be the family of two-edge-colored graphs on k vertices such that one of the colors forms either two disjoint Kk / 2's or simply one Kk / 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 Kn, 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).



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