A uniform version of a theorem by Lindstr\"om

TL;DR

This paper presents a linear algebra-based proof of a uniform version of Lindström's theorem, establishing conditions under which two disjoint set families have equal unions and intersections.

Contribution

It introduces a new uniform version of Lindström's theorem and provides a proof using basic linear algebra techniques.

Findings

Identifies conditions for equal unions and intersections in disjoint set families.

Extends Lindström's theorem to a uniform setting.

Uses linear algebra for the proof approach.

Abstract

We prove the following uniform version of a theorem by Lindstr\"om: Let $\mbox{$\cal F$}:=\{F_i:~ i\in I\}$ be a $k$-uniform set family of $[n]$, where $k\geq 1$. If $|\mbox{$\cal F$}|\geq n+1$, then there exist two disjoint subsets $I_1$ and $I_2$ of $I$ for which $$ \bigcup\limits_{i\in I_1} M_i=\bigcup\limits_{i\in I_2} M_i $$ and $$ \bigcap\limits_{i\in I_1} M_i=\bigcap\limits_{i\in I_2} M_i. $$ Our proof uses basic linear algebra.