A uniform version of a theorem by Lindström

Abstract

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