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- W4378498872 abstract "Let $n,s,$ and $k$ be positive integers such that $kgeq 3$, $sgeq 3$ and $ngeq ks$. An $s$-matching $M_s$ in a $k$-uniform hypergraph is a set of $s$ pairwise disjoint edges. The anti-Ramsey number $textrm{ar}(n,k,M_s)$ 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. In 2013, Ozkahya and Young proposed a conjecture on the exact value of ar$(n,k,M_s)$ for all $n geq sk$ and $k geq 3$. A 2019 result by Frankl and Kupavskii 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$(n,3,M_s)=textrm{ex}(n,3,M_{s-1})+2$. Here $textrm{ex}(n,3,M_{s-1})$ 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 $kgeq 3$, we present a new construction for the lower bound of $textrm{ar}(n,k,M_{s})$ which shows the conjecture by Ozkahya and Young is not true. In particular, for $n=3s$, we prove that $textrm{ar}(n,3,M_s)=textrm{ex}(n,3,M_{s-1})+5$ for sufficiently large $n$." @default.
- W4378498872 created "2023-05-27" @default.
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- W4378498872 date "2023-05-24" @default.
- W4378498872 modified "2023-09-25" @default.
- W4378498872 title "Anti-Ramsey number of matchings in $3$-uniform hypergraphs" @default.
- W4378498872 doi "https://doi.org/10.48550/arxiv.2305.15631" @default.
- W4378498872 hasPublicationYear "2023" @default.
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