Tight bounds for rainbow partial F-tiling in edge-colored complete hypergraphs

For an r-graph F and integers $n,t$ satisfying t ≤ n/v(F), let ar(n,tF) denote the minimum integer N such that every edge-coloring of Kₙʳ 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 Erd{o}s--Simonovits--S\'{o}s~{ESS75}. When F is a single edge, this becomes the rainbow matching problem introduced by Schiermeyer~{Sch04} and \"{O}zkahya--Young~{OY13}. We conduct a systematic study of ar(n,tF) for the case where t is much smaller than ex(n,F)/nʳ⁻¹. Our first main result provides a reduction of ar(n,tF) to ar(n,2F) 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 Tur\'{a}n numbers, determines ar(n,tF) for relatively smaller t. Together, these two results determine ar(n,tF) for a large class of hypergraphs. Additionally, the latter result has the advantage of being applicable to hypergraphs with unknown Tur\'{a}n densities, such as the famous tetrahedron K₄³.