A step towards a general density Corra´di–Hajnal Theorem

For a nondegenerate r-graph F, large n, and t in the regime [0, cF n], where cF > 0 is a constant depending only on F, we present a general approach for determining the maximum number of edges in an n-vertex r-graph that does not contain t + 1 vertex-disjoint copies of F. In fact, our method results in a rainbow version of the above result and includes a characterization of the extremal constructions. Our approach applies to many well-studied hypergraphs (including graphs) such as the edge-critical graphs, the Fano plane, the generalized triangles, hypergraph expansions, the expanded triangles, and hypergraph books. Our results extend old results of Erd˝os, Simonovits, and Moon on complete graphs, and can be viewed as a step towards a general density version of the classical Corr´adi–Hajnal and Hajnal–Szemer´edi Theorems. Our method relies on a novel understanding of the general properties of nondegenerate Tur´an problems, which we refer to as smoothness and boundedness. These properties are satisfied by a broad class of nondegenerate hypergraphs and appear to be worthy of future exploration (Join work with Heng Li, Xizhi Liu, Long-Tu Yuan and Yixiao Zhang).