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?



Maximal number of subsets in $\{1,\dots,n\}$ such that neither is contained in a union of two others
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](http://www.math.uiuc.edu/~z-furedi/PUBS/furedi_erdos_frankl_2cover_free.pdf). See Theorem 2 there. If you look at the [MathSciNet review of the said paper](http://www.ams.org/mathscinet-getitem?mr=677569) 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](http://www.ams.org/mathscinet-getitem?mr=804477) on families not containing sets $A,B\_1,\dotsc,B\_r$ such that $A\subset B\_1\cup\dotsb\cup B\_r$.


