Tight Bounds for Rainbow Partial F -Tiling in Edge-Colored Complete Hypergraphs

For an $r$-graph $F$ and integers $n,t$ satisfying $t\le n/v\left(F\right)$, let $\text{ar}\left(n,tF\right)$ denote the minimum integer $N$ such that every edge-coloring of $K_n^r$ using $N$ colors contains a rainbow copy of $tF$, where $tF$ is the $r$-graphs consisting of $t$ vertex-disjoint copies of $F$. The case $t=1$ is the classical anti-Ramsey problem proposed by Erdős–Simonovits–Sós [1]. When $F$ is a single edge, this becomes the rainbow matching problem introduced by Schiermeyer [2] and Özkahya–Young [3]. We conduct a systematic study of $\text{ar}\left(n,tF\right)$ for the case when $t$ is much smaller than $\text{ex}\left(n,F\right)/n^{r-1}$. Our first main result provides a reduction of $\text{ar}\left(n,tF\right)$ to $\text{ar}\left(n,2F\right)$ when $F$ is bounded and smooth, two properties satisfied by most previously studied hypergraphs. Complementing the first result, the second main result, which utilizes gaps between Turán numbers, determines $\text{ar}\left(n,tF\right)$ for relatively smaller $t$. Together, these two results determine $\text{ar}\left(n,tF\right)$ for a large class of hypergraphs. Additionally, the latter result has the advantage of being applicable to hypergraphs with unknown Turán densities, such as the famous tetrahedron $K_4^3$.