Hypergraph anti-Ramsey theorems

Abstract

The anti-Ramsey number $\mathrm{ar}(n,F)$ 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 $\mathrm{ar}(n,F) = \mathrm{ex}(n,F_{-}) + o(n^r)$ proved by Erd{\H o}s--Simonovits--S{\' o}s, where $F_{-}$ denotes the family of $r$-graphs obtained from $F$ by removing one edge. Second, we determine the exact value of $\mathrm{ar}(n,F)$ for large $n$ in cases where $F$ is the expansion of a specific class of graphs. This extends results of Erd{\H o}s--Simonovits--S{\' o}s on complete graphs to the realm of hypergraphs.