Set systems with few disjoint pairs


Let X = {1,...,n}, and let A be a family of subsets of X. Given the size of A, at least how many pairs of elements of A must be disjoint? In this paper we give a lower bound for the number of disjoint pairs in 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 A satisfies |A|= |X(>r)| then A contains at least as many disjoint pairs as X(>r). The situation is rather different if we restrict our attention to A⊂X(r): 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.

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