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For a property $\Gamma$ and a family of sets $\cF$, let $f(\cF,\Gamma)$ bethe size of the largest subfamily of $\cF$ having property $\Gamma$. For apositive integer $m$, let $f(m,\Gamma)$ be the minimum of $f(\cF,\Gamma)$ overall families of size $m$. A family $\cF$ is said to be $B_d$-free if it has nosubfamily $\cF'=\{F_I: I \subseteq [d]\}$ of $2^d$ distinct sets such that forevery $I,J \subseteq [d]$, 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 $F_1\cup ... F_a \neqF_{a+1}$ whenever $F_1,..,F_{a+1}$ are distinct sets in $\FF$. We verify aconjecture 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 $f(m,a\text{\rm -union free})$.

Large B_d-free and union-free subfamilies