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On stability of rainbow matchings
- Part of
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- Journal:
- Combinatorics, Probability and Computing , First View
- Published online by Cambridge University Press:
- 09 December 2025, pp. 1-14
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- Article
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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 
$t\gt t_0(k)$, 
$\max \{k_1, \ldots , k_{t+1}\}\le k$, and 
$n \gt 2k(t+1)$, the following holds: If 
$F_i$ is a 
$k_i$-uniform hypergraph with vertex set 
$[n]$ and more than 
$ \binom{n}{k_i}-\binom{n-t}{k_i} - \binom{n-t-k}{k_i-1} + 1$ edges 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 
$W\in \binom{[n]}{t}$ such that 
$W$ intersects 
$F_i$ 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 
$n \gt 2k^3t$, generalizing a recent stability result of Frankl and Kupavskii on matchings to rainbow matchings.
