Toward a density Corrádi--Hajnal theorem for degenerate hypergraphs
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| Format: | Preprint |
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2023
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| _version_ | 1866909346170929152 |
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| author | Hou, Jianfeng Hu, Caiyun Li, Heng Liu, Xizhi Yang, Caihong Zhang, Yixiao |
| author_facet | Hou, Jianfeng Hu, Caiyun Li, Heng Liu, Xizhi Yang, Caihong Zhang, Yixiao |
| contents | 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.] |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_15172 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Toward a density Corrádi--Hajnal theorem for degenerate hypergraphs Hou, Jianfeng Hu, Caiyun Li, Heng Liu, Xizhi Yang, Caihong Zhang, Yixiao Combinatorics 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.] |
| title | Toward a density Corrádi--Hajnal theorem for degenerate hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2311.15172 |