Anti-Ramsey Number of Stars in 3-uniform hypergraphs

TL;DR

This paper determines the anti-Ramsey number for k-stars in 3-uniform hypergraphs for larger k and sufficiently large n, extending previous results for smaller stars.

Contribution

It generalizes the anti-Ramsey number for 3-stars to larger sizes, providing exact values for a broader class of hypergraphs.

Findings

Calculated the anti-Ramsey number for 3-stars with size greater than 3.

Extended previous results to larger k and specified n.

Provided exact thresholds for n in relation to k.

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 $ar_r(n,\mathcal{G})$ 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 $ar_3(n,\mathcal{F}_3)$ when $n\geq 20$. In this paper, we determine the anti-Ramsey number $ar_3(n,\mathcal{F}_{k+1})$, where $k\geq 3$ and $n> \frac{5}{2}k^3+\frac{15}{2}k^2+26k-3$.