The generalizations of Erdős matching conjecture for $t$-matching number
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| Format: | Preprint |
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2025
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| _version_ | 1866909740351619072 |
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| author | Zhang, Haixiang Cao, Mengyu Lu, Mei |
| author_facet | Zhang, Haixiang Cao, Mengyu Lu, Mei |
| contents | Define a \textit{$t$-matching} of size $m$ in a $k$-uniform family as a collection $\{A_1, A_2, \ldots, A_m\} \subseteq \binom{[n]}{k}$ such that $|A_i \cap A_j| < t$ for all $1 \leq i < j \leq m$. Let $\mathcal{F}\subseteq \binom{[n]}{k}$. The \textit{$t$-matching number} of $\mathcal{F}$, denoted by $ν_t(\mathcal{F})$, is the maximum size of a $t$-matching contained in $\mathcal{F}$. We study the maximum cardinality of a family $\mathcal{F}\subseteq\binom{[n]}{k}$ with given $t$-matching number, which is a generalization of Erdős matching conjecture, and we additionally prove a stability result. We also determine the second largest maximal structure with $ν_t(\mathcal{F})=s$, extending work of Frankl and Kupavskii \cite{frankl2016two}. Finally, we obtain the extremal $G$-free induced subgraphs of generalized Kneser graph, generalizing Alishahi's results in \cite{alishahi2018extremal}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_12679 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The generalizations of Erdős matching conjecture for $t$-matching number Zhang, Haixiang Cao, Mengyu Lu, Mei Combinatorics 05C35, 05D05, 05D15 Define a \textit{$t$-matching} of size $m$ in a $k$-uniform family as a collection $\{A_1, A_2, \ldots, A_m\} \subseteq \binom{[n]}{k}$ such that $|A_i \cap A_j| < t$ for all $1 \leq i < j \leq m$. Let $\mathcal{F}\subseteq \binom{[n]}{k}$. The \textit{$t$-matching number} of $\mathcal{F}$, denoted by $ν_t(\mathcal{F})$, is the maximum size of a $t$-matching contained in $\mathcal{F}$. We study the maximum cardinality of a family $\mathcal{F}\subseteq\binom{[n]}{k}$ with given $t$-matching number, which is a generalization of Erdős matching conjecture, and we additionally prove a stability result. We also determine the second largest maximal structure with $ν_t(\mathcal{F})=s$, extending work of Frankl and Kupavskii \cite{frankl2016two}. Finally, we obtain the extremal $G$-free induced subgraphs of generalized Kneser graph, generalizing Alishahi's results in \cite{alishahi2018extremal}. |
| title | The generalizations of Erdős matching conjecture for $t$-matching number |
| topic | Combinatorics 05C35, 05D05, 05D15 |
| url | https://arxiv.org/abs/2508.12679 |