A step towards a general density Corrádi-Hajnal Theorem

For a nondegenerate r-graph F, large n, and t in the regime [0, CFn], 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 Erdos [12], Simonovits [76], and Moon [58] on complete graphs, and can be viewed as a step towards a general density version of the classical Corr di Hajnal [10] and Hajnal Szemer di [32] Theorems. Our method relies on a novel understanding of the general properties of nondegenerate Tur n 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.