Erdős–Hajnal conjecture for rainbow triangles

From papers

Let rr and tt be positive integers, and let r(t)r(t) denote the two-color Ramsey number: the least integer mm such that every 22-coloring of the edges of the complete graph on mm vertices contains a monochromatic clique of order tt. Define

N(r,t)=(r(t)1)r/2N(r,t)=\left(r(t)-1\right)^{r/2}

when rr is even, and

N(r,t)=(t1)(r(t)1)(r1)/2N(r,t)=(t-1)\left(r(t)-1\right)^{(r-1)/2}

when rr 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 KtK_t is a complete subgraph on tt vertices whose edges all have one color. Erdős–Hajnal conjecture. For n>N(r,t)n>N(r,t), every rr-coloring of the edges of the complete graph on nn vertices has a rainbow triangle or a monochromatic KtK_t.

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 t=3t=3; the general case remains open in the supplied source.

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Sources & referencesView supporting material

Primary source

J. Fox, A. Grinshpun and J. Pach, “The Erdős-Hajnal conjecture for rainbow triangles”, arXiv:1303.2951 (2013).

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