Turán’s theorem with colors
نویسندگان
چکیده
We consider a generalization of Turán’s theorem for edge-colored graphs. Suppose that R (red) and B (blue) are graphs on the same vertex set of size n. We conjecture that if R and B each have more than (1− 1/k)n/2 edges, and K is a (k+ 1)-clique whose edges are arbitrarily colored with red and blue, then R ∪B contains a colored copy of K, for all k + 1 6∈ {4, 6, 8}. If k + 1 ∈ {4, 6, 8}, then the same conclusion holds except for certain specified edge-colorings of Kk+1. We prove this conjecture for all 2-edge-colorings of Kk+1 that contain a monochromatic Kk. We also prove the conjecture for k + 1 ∈ {3, 4, 5}.
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