# Set systems with few disjoint pairs

## Abstract

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.

## Publication Title

Combinatorica

## Recommended Citation

Bollobás, B., & Leader, I.
(2003). Set systems with few disjoint pairs.* Combinatorica**, 23* (4), 559-570.
https://doi.org/10.1007/s00493-003-0033-0