On sets not belonging to algebras and rainbow matchings in graphs

TL;DR

This paper investigates a combinatorial problem involving equivalence relations and rainbow matchings, proving an asymptotic answer to a question posed by Grinblat and improving bounds on the minimal number needed.

Contribution

The authors confirm Grinblat's conjecture asymptotically, showing that the minimal number is close to 3n, refining previous bounds significantly.

Findings

Established that n+o(n) suffices for rainbow matchings

Improved the upper bound from 16n/5 + O(1) to asymptotically 3n

Connected the problem to edge-colored multigraphs

Abstract

Motivated by a question of Grinblat, we study the minimal number $\mathfrak{v}(n)$ that satisfies the following. If $A_1,\ldots, A_n$ are equivalence relations on a set $X$ such that for every $i\in[n]$ there are at least $\mathfrak{v}(n)$ elements whose equivalence classes with respect to $A_i$ are nontrivial, then $A_1, \ldots, A_n$ contain a rainbow matching, i.e. there exist $2n$ distinct elements $x_1,y_1,\ldots,x_n,y_n\in X$ with $x_i\sim_{A_i} y_i$ for each $i\in [n]$. Grinblat asked whether $\mathfrak{v}(n) = 3n-2$ for every $n\geq 4$. The best-known upper bound was $\mathfrak{v}(n) \leq 16n/5 + \mathcal{O}(1)$ due to Nivash and Omri. Transferring the problem into the setting of edge-coloured multigraphs, we affirm Grinblat's question asymptotically, i.e. we show that $\mathfrak{v}(n) = 3n+o(n)$.