Anti-Ramsey number of matchings in 3-uniform hypergraphs

地   点：  腾讯会议 APP()

报告人：  鲁红亮

单   位：  西安交通大学

邀请人：  汪彦

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报告摘要：  

Let $n,s,$ and $k$ be positive integers such that $k\geq 3$, $s\geq 3$ and $n\geq ks$. An $s$-matching $M_s$ in a $k$-uniform hypergraph is a set of $s$ pairwise disjoint edges. The anti-Ramsey number $\ar(n,k,M_s)$ of an $s$-matching  is the smallest integer $c$ such that each edge-coloring of the $n$-vertex $k$-uniform complete hypergraph with exactly $c$ colors contains an $s$-matching  with distinct colors.  The value of $\ar(n,k,M_s)$ was conjectured by \"Ozkahya and Young (Anti-Ramsey number of matchings in hypergraphs, {\it Discrete Math.}, {\bf 313} (2013), 2359--2364) for all  $n \geq sk$ and $k \geq 3$. Frankl and Kupavskii (Two problems on matchings in set families - in the footsteps of Erd\H os and Kleitman  verified this conjecture for all $n \geq sk+(s-1)(k-1)$ and $k \geq 3$. We aim to determine the value of $\ar(n,3,M_s)$ for $3s \leq n < 5s-2$ in this paper. Namely,  we prove that if $3s<n<5s-2$ and $n$ is large enough, then $\ar(n,3,M_s)=\ex(n,3,M_{s-1})+2$. Here  $\ex(n,3,M_{s-1})$ is the Tur\'an number of an $(s-1)$-matching. Thus this result confirms the conjecture of \"Ozkahya and Young for $k=3$, $3s<n<5s-2$ and sufficiently large $n$. For $n=ks$ and $k\geq 3$, we present a new construction for the lower bound of $\ar(n,k,M_{s})$ which shows the conjecture by \"Ozkahya and Young is not true. In particular, for $n=3s$, we prove that $\ar(n,3,M_s)=\ex(n,3,M_{s-1})+5$ for sufficiently large $n$.

This joint work with Mingyang Guo and Xing Peng.