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?