Anti-Ramsey Number of Stars in 3-uniform hypergraphs

Abstract

An edge-colored hypergraph is called \emph{a rainbow hypergraph} if all the colors on its edges are distinct. Given two positive integers $n,r$ and an $r$-uniform hypergraph $\mathcal{G}$, the anti-Ramsey number $a{r}_r(n,\mathcal{G})$ is defined to be the minimum number of colors $t$ such that there exists a rainbow copy of $\mathcal{G}$ in any exactly $t$-edge-coloring of the complete $r$-uniform hypergraph of order $n$. Let ${\mathcal{F}}_k$ denote the 3-graph ($k$-star) consisting of $k$ edges sharing exactly one vertex. Tang, Li and Yan \cite{YTG} determined the value of $a{r}_3(n,{\mathcal{F}}_3)$ when $n\ge 20$. In this paper, we determine the anti-Ramsey number $a{r}_3(n,{\mathcal{F}}_{k+1})$, where $k\ge 3$ and $n>\frac{5}{2}{k}^3+\frac{15}{2}{k}^2+26k-3$.