Hypergraph Anti-Ramsey Theorems

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