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arXiv · 1607.06600

High girth hypergraphs with unavoidable monochromatic or rainbow edges

Maria Axenovich·Annette Karrer
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Abstract

A classical result of Erd\H{o}s and Hajnal claims that for any integers $k, r, g \geq 2$ there is an $r$-uniform hypergraph of girth at least $g$ with chromatic number at least $k$. This implies that there are sparse hypergraphs such that in any coloring of their vertices with at most $k-1$ colors there is a monochromatic hyperedge. We show that for any integers $r, g\geq 2$ there is an $r$-uniform hypergraph of girth at least $g$ such that in any coloring of its vertices there is either a monochromatic or a rainbow (totally multicolored) edge. We give a probabilistic and a deterministic proof of this result.

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Maria Axenovich, Annette Karrer. 2016-07-22. High girth hypergraphs with unavoidable monochromatic or rainbow edges. https://arxiv.org/abs/1607.06600

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