TITLE: Maximal number of subsets in $\{1,\dots,n\}$ such that neither is contained in a union of two others
QUESTION [16 upvotes]: What are known estimates for maximal $M$ for which their exists subsets $A_1,\dots,A_M$ in $\{1,\dots,n\}$ such that there do not exist different indexes $i,j,k$ for which $A_i\subset A_j\cup A_k$?
It is not hard to prove some exponential bounds $c_1^n<M<c_2^n$ for some $1<c_1<c_2<2$, but maybe sharp exponent is known?

REPLY [6 votes]: To best of my knowledge, the only bounds on this problem in the literature are in the the old paper of Erdős, Frankl, and Füredi. See Theorem 2 there. If you look at the MathSciNet review of the said paper you will find papers that referenced this paper. Perhaps, some might be of use to you.
There is also a follow-up paper by the same authors on families not containing sets $A,B_1,\dotsc,B_r$ such that $A\subset B_1\cup\dotsb\cup B_r$.