Erdős–Hajnal conjecture for rainbow triangles
Erdős–Hajnal conjecture for rainbow triangles
Let and be positive integers, and let denote the two-color Ramsey number: the least integer such that every -coloring of the edges of the complete graph on vertices contains a monochromatic clique of order . Define
when is even, and
when is odd. A coloring of the edges of a complete graph is a rainbow triangle if some triangle has three differently colored edges, and a monochromatic is a complete subgraph on vertices whose edges all have one color. Erdős–Hajnal conjecture. For , every -coloring of the edges of the complete graph on vertices has a rainbow triangle or a monochromatic .
This conjecture asserts that the lexicographic-product constructions described in the paper give the best possible bounds for avoiding both rainbow triangles and monochromatic cliques. It was verified by Chung and Graham in the case ; the general case remains open in the supplied source.
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Primary source
J. Fox, A. Grinshpun and J. Pach, “The Erdős-Hajnal conjecture for rainbow triangles”, arXiv:1303.2951 (2013).
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