Improved Bounds on Rainbow $k$-partite Matchings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911100673458176 |
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| author | Saengrungkongka, Pitchayut |
| author_facet | Saengrungkongka, Pitchayut |
| contents | Let $n$, $s$, and $k$ be positive integers. We say that a sequence $f_1,\dots,f_s$ of nonnegative integers is satisfying if for any collection of $s$ families $\mathcal F_1,\dots,\mathcal F_s\subseteq [n]^k$ such that $|\mathcal F_i|=f_i$ for all $i$, there exists a rainbow matching, i.e., a list of pairwise disjoint tuples $F_1\in\mathcal F_1$, $\dots$, $F_s\in\mathcal F_s$. We investigate the question, posed by Kupavskii and Popova, of determining the smallest $c=c(n,s,k)$ such that the arithmetic progression $c$, $n^{k-1}+c$, $2n^{k-1}+c$, $\dots$, $(s-1)n^{k-1}+c$ is satisfying. We prove that the sequence is satisfying for $c=Ω_k(\max(s^2n^{k-2}, sn^{k-3/2}\sqrt{\log s}))$, improving the previous result by Kupavskii and Popova. We also study satisfying sequences for $k=2$ using the polynomial method, extending the previous result by Kupavskii and Popova to when $n$ is not prime. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_07331 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Improved Bounds on Rainbow $k$-partite Matchings Saengrungkongka, Pitchayut Combinatorics 05D05, 05D40 Let $n$, $s$, and $k$ be positive integers. We say that a sequence $f_1,\dots,f_s$ of nonnegative integers is satisfying if for any collection of $s$ families $\mathcal F_1,\dots,\mathcal F_s\subseteq [n]^k$ such that $|\mathcal F_i|=f_i$ for all $i$, there exists a rainbow matching, i.e., a list of pairwise disjoint tuples $F_1\in\mathcal F_1$, $\dots$, $F_s\in\mathcal F_s$. We investigate the question, posed by Kupavskii and Popova, of determining the smallest $c=c(n,s,k)$ such that the arithmetic progression $c$, $n^{k-1}+c$, $2n^{k-1}+c$, $\dots$, $(s-1)n^{k-1}+c$ is satisfying. We prove that the sequence is satisfying for $c=Ω_k(\max(s^2n^{k-2}, sn^{k-3/2}\sqrt{\log s}))$, improving the previous result by Kupavskii and Popova. We also study satisfying sequences for $k=2$ using the polynomial method, extending the previous result by Kupavskii and Popova to when $n$ is not prime. |
| title | Improved Bounds on Rainbow $k$-partite Matchings |
| topic | Combinatorics 05D05, 05D40 |
| url | https://arxiv.org/abs/2508.07331 |