Hypergraph Anti-Ramsey Theorems

Abstract

The anti-Ramsey number $\text{ar}\left(n,F\right)$ of an $r$-graph $F$ is the minimum number of colors needed to color the complete $n$-vertex $r$-graph to ensure the existence of a rainbow copy of $F$. We establish a removal-type result for the anti-Ramsey problem of $F$ when $F$ is the expansion of a hypergraph with a smaller uniformity. We present two applications of this result. First, we refine the general bound $\text{ar}\left(n,F\right)=\text{ex}\left(n,F_{-}\right)+o\left(n^r\right)$ proved by Erdős–Simonovits–Sós, where $F_{-}$ denotes the family of $r$-graphs obtained from $F$ by removing one edge. Second, we determine the exact value of $\text{ar}\left(n,F\right)$ for large $n$ in cases where $F$ is the expansion of a specific class of graphs. This extends results of Erdős–Simonovits–Sós on complete graphs to the realm of hypergraphs.