High girth hypergraphs with unavoidable monochromatic or rainbow edges

DOI10.7151/DMGT.2291arXiv1607.06600WikidataQ126309619 ScholiaQ126309619MaRDI QIDQ6275829

Maria Axenovich, Annette Karrer

Publication date: 22 July 2016

Abstract: A classical result of ErdH{o}s and Hajnal claims that for any integers $k,r,ggeq2$ 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 $r,ggeq2$ 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.

Mathematics Subject Classification ID

Extremal problems in graph theory (05C35) Hypergraphs (05C65) Coloring of graphs and hypergraphs (05C15)

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