Toward a density Corrádi--Hajnal theorem for degenerate hypergraphs

Abstract

Given an $r$-graph $F$ with $r\ge 2$, let $\mathrm{ex}(n,(t+1)F)$ 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 [J. Hou, H. Li, X. Liu, L.-T. Yuan, and Y. Zhang. A step towards a general density Corrádi--Hajnal theorem. arXiv:2302.09849, 2023.] on nondegenerate hypergraphs, we initiate a systematic study on $\mathrm{ex}(n,(t+1)F)$ for degenerate hypergraphs $F$. For a broad class of degenerate hypergraphs $F$, we present near-optimal upper bounds for $\mathrm{ex}(n,(t+1)F)$ when $n$ is sufficiently large and $t$ lies in intervals $\left[0,\frac{\epsilon \cdot \mathrm{ex}(n,F)}{n^{r-1}}\right]$, $\left[\frac{\mathrm{ex}(n,F)}{\epsilon n^{r-1}},\epsilon n\right]$, and $\left[(1-\epsilon )\frac{n}{v(F)},\frac{n}{v(F)}\right]$, 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, for graphs, we offer a characterization of extremal constructions within the second interval. 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 [J. Hou, H. Li, X. Liu, L.-T. Yuan, and Y. Zhang. A step towards a general density Corrádi--Hajnal theorem. arXiv:2302.09849, 2023.]

fixed Proposition 2.11