Exact solution of some Turán-type problems (Q1112819)

Fifteen years ago Chvátal conjectued that if \({\mathcal F}\) is a family of k subsets of an n-set, \(| {\mathcal F}| >\left( \begin{matrix} n-1\\ k- 1\end{matrix} \right)\), d is an arbitrary integer with \(d\leq k-1\) and \((d+1)k\leq dn\), then there exist \(d+1\) sets in \({\mathcal F}\) with empty intersection such that the intersection of any d of them is non-empty. The validity of this conjecture is established for \(n\geq n_ 0(k)\), in a more general framework. Another problem which is solved asymptotically is when the excluded configuration is a fixed sunflower.