Toward a density Corrádi–Hajnal theorem for degenerate hypergraphs

Abstract

Given an r-graph F with $r\ge 2$, let $\mathrm{ex}(n,(t+1\left)F\right)$ denote the maximum number of edges in an n-vertex r-graph with at most t pairwise vertex-disjoint copies of F. Extending several old results and complementing prior work [34] on nondegenerate hypergraphs, we initiate a systematic study on $\mathrm{ex}(n,(t+1\left)F\right)$ for degenerate hypergraphs F.

For a broad class of degenerate hypergraphs F, we present near-optimal upper bounds for $\mathrm{ex}(n,(t+1\left)F\right)$ when n is sufficiently large and t lies in intervals $[0,\frac{\epsilon \cdot \mathrm{ex}(n,F)}{n^{r-1}}]$, $[\frac{\mathrm{ex}(n,F)}{\epsilon n^{r-1}},\epsilon n]$, and $\left[\right(1-\epsilon )\frac{n}{v\left(F\right)},\frac{n}{v\left(F\right)}]$, where $\epsilon >0$ is a constant depending only on F. Our results reveal very different structures for extremal constructions across the three intervals, and we provide characterizations of extremal constructions within the first interval. Additionally, we characterize extremal constructions within the second interval for graphs. Our proof for the first interval also applies to a special class of nondegenerate hypergraphs, including those with undetermined Turán densities, partially improving a result in [34].