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 K₍^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 Erdos--Simonovits--S\'os~ESS75. When F is a single edge, this becomes the rainbow matching problem introduced by Schiermeyer~Sch04 and \"Ozkahya--Young~OY13. We conduct a systematic study of ar (n, tF) for the case where t is much smaller than ex (n, F) /n^r-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\'an 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\'an densities, such as the famous tetrahedron K₄^3.