Fifteen years ago Chvátal conjectured that if F is a family of k subsets of an n-set, |F|>( n-1 k-1), d is an arbitrary integer with d ≤ k - 1 and (d + 1) k ≤ dn, then there exist d + 1 sets in F with empty intersection such that the intersection of any d of them is non-empty. The validity of this conjecture is established for n ≥ n0(k), in a more general framework. Another problem which is solved asymptotically is when the excluded configuration is a fixed sunflower.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics