Set Systems with few Disjoint Pairs

Let X = {1, . . . , n}, and let ${\user1{A}}$ be a family of subsets of X. Given the size of ${\user1{A}}$, at least how many pairs of elements of ${\user1{A}}$ must be disjoint? In this paper we give a lower bound for the number of disjoint pairs in ${\user1{A}}$. The bound we obtain is essentially best possible. In particular, we give a new proof of a result of Frankl and of Ahlswede, that if ${\user1{A}}$ satisfies ${\left| {\user1{A}} \right|} = {\left| {X^{{{\left( { \geqslant r} \right)}}} } \right|}$ then ${\user1{A}}$ contains at least as many disjoint pairs as X ^(≥r).

The situation is rather different if we restrict our attention to ${\user1{A}} \subset X^{{{\left( r \right)}}}$: then we are asking for the minimum number of edges spanned by a subset of the Kneser graph of given size. We make a conjecture on this lower bound, and disprove a related conjecture of Poljak and Tuza on the largest bipartite subgraph of the Kneser graph.