Extremal families containing no two sets and their union

The family F⊆2[n] of subsets of [n]={1,2,…,n} is 2-union free if it does not contain three different sets A, B, C such that A=B∪C, and F⊆2[n] is union free if it does not contain k+1⩾3 different sets A0, A1, …, Ak such that . Let cn, bn be the maximum size of a 2-union free, union free family, respectively. Simple construction shows that , which slightly improves the lower bound obtained by Kleitman if n is odd. Another more complicated construction further improves this lower bound. Union free family of m subsets of [n] naturally corresponds to an m×n (0,1) matrix with independent rows. The result is related to maximum number m of independent rows in some (0,1) matrix with n columns.