Rainbow matchings and algebras of sets

TL;DR

This paper investigates the minimum conditions needed for the existence of rainbow matchings in algebras of sets, improving the upper bound on this threshold for large n.

Contribution

It provides a tighter upper bound on the minimum number of elements required to guarantee rainbow matchings in the context of equivalence relations.

Findings

Improved upper bound on v(n) to 16n/5 + O(1) for large n.

Addresses Grinblat's question on the exact value of v(n).

Advances understanding of combinatorial structures in algebras of sets.

Abstract

Grinblat (2002) asks the following question in the context of algebras of sets: What is the smallest number $\mathfrak v = \mathfrak v(n)$ such that, if $A_1, \ldots, A_n$ are $n$ equivalence relations on a common finite ground set $X$, such that for each $i$ there are at least $\mathfrak v$ elements of $X$ that belong to $A_i$-equivalence classes of size larger than $1$, then $X$ has a rainbow matching---a set of $2n$ distinct elements $a_1, b_1, \ldots, a_n, b_n$, such that $a_i$ is $A_i$-equivalent to $b_i$ for each $i$? Grinblat has shown that $\mathfrak v(n) \le 10n/3 + O(\sqrt{n})$. He asks whether $\mathfrak v(n) = 3n-2$ for all $n\ge 4$. In this paper we improve the upper bound (for all large enough $n$) to $\mathfrak v(n) \le 16n/5 + O(1)$.