Abstract
Let F be a family of finite sets. A subfamily F' of F is B2-free if it does not contain distinct sets A1,...,A4 such that the union of A1 and A2 is A3 and the intersection of A1 and A2 is A4.
Similarly, a subfamily is Bd-free if it does not contain 2^d sets that form a copy of the Boolean lattice Bd. The size of the largest Bd-free subfamily of F will be denoted by f(F,Bd-free), and f(m,Bd-free) will denote the minimum of f(F,Bd-free) over all families F of cardinality m.
Instead of ''Bd-free" we can take any other property Gamma, and define f(m,Gamma) analogously. We provide lower and upper bounds for f(m,Bd-free).
In particular, we show that f(m,B2-free) is at least cm^{2/3} for a constant c, verifying conjecture by Erdos and Shelah from 1972. We also generalize the notion of union-free families investigated by Erdos and Komlos, Kleitman, and Erdos and Shelah, and prove results about a-union-free families, i.e., families that do not contain distinct sets A1,...,A{a+1} such that the union of A1, A2, ... and Aa is A{a+1}.