Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Erd\H{o}s-Hajnal conjecture for rainbow triangles   We prove that every 3-coloring of the edges of the complete graph on n
vertices without a rainbow triangle contains a set of order Omega(n^{1/3}log^2
n) which uses at most two colors, and this bound is tight up to a constant
factor. This verifies a conjecture of Hajnal which is a case of the multicolor
generalization of the well-known Erd\H{o}s-Hajnal conjecture. We further
establish a generalization of this result. For fixed positive integers s and r
with s at most r, we determine a constant c_{r,s} such that the following
holds. Every r-coloring of the edges of the complete graph on n vertices
without a rainbow triangle contains a set of order Omega(n^{r(r-1)/s(s-1)}(\log
n)^{c_{r,s}}) which uses at most s colors, and this bound is tight apart from
the implied constant factor. The proof of the lower bound utilizes Gallai's
classification of rainbow-triangle free edge-colorings of the complete graph, a
new weighted extension of Ramsey's theorem, and a discrepancy inequality in
edge-weighted graphs. The proof of the upper bound uses Erd\H{o}s' lower bound
on Ramsey numbers by considering lexicographic products of 2-edge-colorings of
complete graphs without large monochromatic cliques.

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The categories of this paper are: bayes-an
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: cs
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The categories of this paper are: comp-gas
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The categories of this paper are: quant-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: stat
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The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: math-ph
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The categories of this paper are: adap-org
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The categories of this paper are: hep-ph
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The categories of this paper are: q-bio
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin