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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2302.06146 |
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| _version_ | 1866914347713822720 |
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| author | Lu, Hongliang Wang, Yan Yu, Xingxing |
| author_facet | Lu, Hongliang Wang, Yan Yu, Xingxing |
| contents | We show that for any integer $k\ge 1$ there exists an integer $t_0(k)$ such that for integers $t, k_1, \ldots, k_{t+1}, n$ with $t>t_0(k)$, $\max\{k_1, \ldots, k_{t+1}\}\le k$, and $n > 2k(t+1)$, the following holds: If $F_i \subseteq {[n]\choose k_i}$ and $|F_i|> {n\choose k_i}-{n-t\choose k_i} - {n-t-k \choose k_i-1} + 1$ for all $i \in [t+1]$, then either $\{F_1,\ldots, F_{t+1}\}$ admits a rainbow matching of size $t+1$ or there exists $W\in {[n]\choose t}$ such that $W$ is a vertex cover of $F_i$ for all $i\in [t+1]$. This may be viewed as a rainbow non-uniform extension of the classical Hilton-Milner theorem. We also show that the same holds for every $t$ and $n > 2k^3t$, generalizing a recent stability result of Frankl and Kupavskii on matchings to rainbow matchings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_06146 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On stability of rainbow matchings Lu, Hongliang Wang, Yan Yu, Xingxing Combinatorics We show that for any integer $k\ge 1$ there exists an integer $t_0(k)$ such that for integers $t, k_1, \ldots, k_{t+1}, n$ with $t>t_0(k)$, $\max\{k_1, \ldots, k_{t+1}\}\le k$, and $n > 2k(t+1)$, the following holds: If $F_i \subseteq {[n]\choose k_i}$ and $|F_i|> {n\choose k_i}-{n-t\choose k_i} - {n-t-k \choose k_i-1} + 1$ for all $i \in [t+1]$, then either $\{F_1,\ldots, F_{t+1}\}$ admits a rainbow matching of size $t+1$ or there exists $W\in {[n]\choose t}$ such that $W$ is a vertex cover of $F_i$ for all $i\in [t+1]$. This may be viewed as a rainbow non-uniform extension of the classical Hilton-Milner theorem. We also show that the same holds for every $t$ and $n > 2k^3t$, generalizing a recent stability result of Frankl and Kupavskii on matchings to rainbow matchings. |
| title | On stability of rainbow matchings |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2302.06146 |