A step towards a general density Corr\'{a}di--Hajnal Theorem

TL;DR

This paper develops a general method to determine the maximum edges in large hypergraphs avoiding multiple disjoint copies of a fixed hypergraph, extending classical results and including a rainbow version with extremal characterizations.

Contribution

It introduces a unified approach for hypergraph Turán-type problems, generalizing classical theorems and applying to various well-studied hypergraphs with extremal constructions.

Findings

Provides a general approach for hypergraph extremal problems.

Includes a rainbow version and extremal characterization.

Extends classical results to a broader hypergraph setting.

Abstract

For a nondegenerate $r$-graph $F$, large $n$, and $t$ in the regime $[0, c_{F} n]$, where $c_F>0$ 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 Simonovits~\cite{SI68} and Moon~\cite{Moon68} on complete graphs and can be viewed as a step towards a general density version of the classical Corr\'{a}di--Hajnal Theorem~\cite{CH63}.