Problem setting
Consider a set $ S = \big\{ 1,2,\cdots,n \big\}$. Now consider $k$ equal-sized subsets $S_i \subset S$ s.t of size $\big|S_i\big|=n' \;\forall i$.
Consider a $k\times k$ matrix $M$ s.t $\;M_{ij} = \big|S_i\cap S_j\big|\big/n'$, size of the intersection of the $i^{th}, j^{th}$ subset divided by $n'$.
Question
Does $M_{ij}$ lie in the polytope of zero-one matrices? Paraphrasing the question - Can we express every $M_{ij}$ as a convex combination of zero-one matrices?