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arXiv · 1901.06092

Anti-Ramsey numbers of paths and cycles in hypergraphs

Ran Gu·Jiaao Li·Yongtang Shi
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Abstract

The anti-Ramsey problem was introduced by Erdős, Simonovits and Sós in 1970s. The anti-Ramsey number of a hypergraph $\mathcal{H}$, $ar(n,s, \mathcal{H})$, is the smallest integer $c$ such that in any coloring of the edges of the $s$-uniform complete hypergraph on $n$ vertices with exactly $c$ colors, there is a copy of $\mathcal{H}$ whose edges have distinct colors. In this paper, we determine the anti-Ramsey numbers of linear paths and loose paths in hypergraphs for sufficiently large $n$, and give bounds for the anti-Ramsey numbers of Berge paths. Similar exact anti-Ramsey numbers are obtained for linear/loose cycles, and bounds are obtained for Berge cycles. Our main tools are path extension technique and stability results on hypergraph Turán problems of paths and cycles.

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Ran Gu, Jiaao Li, Yongtang Shi. 2019-11-12. Anti-Ramsey numbers of paths and cycles in hypergraphs. https://arxiv.org/abs/1901.06092

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