Anti-Ramsey numbers of loose paths and cycles in uniform hypergraphs

TL;DR

This paper determines the exact anti-Ramsey numbers for loose paths and cycles in uniform hypergraphs for all sufficiently large parameters, advancing understanding of rainbow structures in hypergraph colorings.

Contribution

It provides the first exact values of anti-Ramsey numbers for loose paths and cycles in r-uniform hypergraphs for all k ≥ 4 and r ≥ 3.

Findings

Exact anti-Ramsey numbers for loose paths and cycles are established.

Results apply to all k ≥ 4 and r ≥ 3, covering broad cases.

Advances the theory of rainbow hypergraph structures.

Abstract

For a fixed family of $r$-uniform hypergraphs $\mathcal{F}$, the anti-Ramsey number of $\mathcal{F}$, denoted by $ ar(n,r,\mathcal{F})$, is the minimum number $c$ of colors such that for any edge-coloring of the complete $r$-uniform hypergraph on $n$ vertices with at least $c$ colors, there is a rainbow copy of some hypergraph in $\mathcal{F}$. Here, a rainbow hypergraph is an edge-colored hypergraph with all edges colored differently. Let $\mathcal{P}_k$ and $\mathcal{C}_k$ be the families of loose paths and loose cycles with $k$ edges in an $r$-uniform hypergraph, respectively. In this paper, we determine the exact values of $ ar(n,r,\mathcal{P}_k)$ and $ ar(n,r,\mathcal{C}_k)$ for all $k\geq 4$ and $r\geq 3$.