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arXiv · 2311.15172

Toward a density Corrádi--Hajnal theorem for degenerate hypergraphs

Jianfeng Hou·Caiyun Hu·Heng Li·Xizhi Liu·Caihong Yang·Yixiao Zhang
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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 $\left[0, \frac{\varepsilon \cdot \mathrm{ex}(n,F)}{n^{r-1}}\right]$, $\left[\frac{\mathrm{ex}(n,F)}{\varepsilon n^{r-1}}, \varepsilon n \right]$, and $\left[ (1-\varepsilon)\frac{n}{v(F)}, \frac{n}{v(F)} \right]$, where $\varepsilon > 0$ 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.]

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Jianfeng Hou, Caiyun Hu, Heng Li, Xizhi Liu, Caihong Yang, Yixiao Zhang. 2024-10-11. Toward a density Corrádi--Hajnal theorem for degenerate hypergraphs. https://arxiv.org/abs/2311.15172

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