# Anti-Ramsey numbers of paths and cycles in hypergraphs

## TL;DR

This paper determines the anti-Ramsey numbers for various paths and cycles in hypergraphs, providing exact values for large n and bounds for Berge paths and cycles, using path extension and stability methods.

## Contribution

It establishes exact anti-Ramsey numbers for linear and loose paths and cycles in hypergraphs, and bounds for Berge paths and cycles, advancing hypergraph coloring theory.

## Key findings

- Exact anti-Ramsey numbers for linear and loose paths and cycles.
- Bounds for anti-Ramsey numbers of Berge paths and cycles.
- Application of path extension and stability techniques.

## Abstract

The anti-Ramsey problem was introduced by Erd\H{o}s, Simonovits and S\'{o}s in 1970s. The anti-Ramsey number of a hypergraph $\mathcal{H}$, $ar(n,s, \mathcal{H})$, is the smallest integer $c$ such that in any coloring of the edges of the $s$-uniform complete hypergraph on $n$ vertices with exactly $c$ colors, there is a copy of $\mathcal{H}$ whose edges have distinct colors. In this paper, we determine the anti-Ramsey numbers of linear paths and loose paths in hypergraphs for sufficiently large $n$, and give bounds for the anti-Ramsey numbers of Berge paths. Similar exact anti-Ramsey numbers are obtained for linear/loose cycles, and bounds are obtained for Berge cycles. Our main tools are path extension technique and stability results on hypergraph Tur\'{a}n problems of paths and cycles.