Large B_d-free and union-free subfamilies

TL;DR

This paper investigates the maximum size of large subfamilies with specific combinatorial properties, confirming a conjecture for B_2-free families and providing bounds for other related properties.

Contribution

It proves a conjecture on B_2-free families and establishes bounds for B_d-free and a-union free families, advancing understanding of set family structures.

Findings

Confirmed Erdős and Shelah's conjecture for B_2-free families.

Derived bounds for f(m, B_d-free) and f(m, a-union free).

Provided new insights into the structure of large set families.

Abstract

For a property $\Gamma$ and a family of sets $\cF$, let $f(\cF,\Gamma)$ be the size of the largest subfamily of $\cF$ having property $\Gamma$. For a positive integer $m$, let $f(m,\Gamma)$ be the minimum of $f(\cF,\Gamma)$ over all families of size $m$. A family $\cF$ is said to be $B_d$-free if it has no subfamily $\cF'=\{F_I: I \subseteq [d]\}$ of $2^d$ distinct sets such that for every $I,J \subseteq [d]$, both $F_I \cup F_J=F_{I \cup J}$ and $F_I \cap F_J = F_{I \cap J}$ hold. A family $\cF$ is $a$-union free if $F_1\cup ... F_a \neq F_{a+1}$ whenever $F_1,..,F_{a+1}$ are distinct sets in $\FF$. We verify a conjecture of Erd\H os and Shelah that $f(m, B_2\text{\rm -free})=\Theta(m^{2/3})$. We also obtain lower and upper bounds for $f(m, B_d\text{\rm -free})$ and $f(m,a\text{\rm -union free})$.