# Existence of disjoint subsets of a family of subsets such that each element appears the same number of times in each

Let $$A$$ be a set with $$n$$ elements. Consider a family $$B$$ of subsets of $$A$$ i.e. $$B\subseteq\mathcal{P}(A)$$.

How large must $$B$$ be to guarantee the existence of two nonempty disjoint subsets $$X,Y\subseteq B$$ (i.e. $$X\cap Y=\emptyset$$) such that $$\forall a\in A$$, the number of elements of $$X$$ that contain $$a$$ is equal to the number of elements of $$Y$$ that contain $$a$$?

Notice that your question is equivalent to asking: for a given $$n$$, what is the smallest $$m$$ such that any matrix $$M\in\{0,1\}^{n\times m}$$ has two disjoint subsets of the columns which have the same sum. The answer to your question is $$m=\Theta(n\log n)$$, which was found by Erdos and Renyi (they actually consider fixing $$m$$ and maximizing $$n$$, but this yields the same answer asyptotically). You can find their proof in the paper On two problems of information theory'' https://users.renyi.hu/~p_erdos/1963-12.pdf