I'm looking for the answer of following problem: Find the maximal size of $S\subseteq 2^{[n]}$ such that there are no distinct $A,B,C\in S$ such that $A\cup B\not = C$. A possible construction is all set of
odd size, which $\#S=2^{n-1}$. A possible construction is all set of
size $\lfloor n/2\rfloor$, which $\#S={n\choose {\lfloor n/2\rfloor}}$. Is this maximal?
I've found https://arxiv.org/pdf/1601.03659.pdf. There is reference in that which Kleitman give an upper bound $(1+o(1)){n\choose {\lfloor n/2\rfloor}}$. Seemingly that's an open problem for an exact number.