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A uniform version of a theorem by Lindström

Gábor Hegedüs

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 $k\geq 1$. 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.

A uniform version of a theorem by Lindström

Abstract

We prove the following uniform version of a theorem by Lindström: Let \cal F be a -uniform set family of , where . If \cal F, then there exist two disjoint subsets and of for which and Our proof uses basic linear algebra.
Paper Structure (3 sections, 4 theorems, 8 equations)

This paper contains 3 sections, 4 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. The proof of our main result
  3. Concluding remark

Key Result

Theorem 1.1

Let $\hbox{$\cal F$}$ be a set family on $[n]$. If $|\hbox{$\cal F$}|\geq n+1$, then $\hbox{$\cal F$}$ is a $\cup$-balanced family.

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Conjecture 1