The structure of intersecting families of sets

John Talbot
Oxford


The Erdos-Ko-Rado theorem tells us how large an intersecting family of r-sets from an n-set can be, while a result due to Lovasz gives bounds on the number of singletons that can occur as intersections of sets from such a family.

After giving a gentle introduction to this area we consider a more general problem. Given an intersecting family of r-sets from an n-set how many k-sets can occur as intersections of sets from the family? For k=r and k=1 this reduces to the problems described above. We solve some new cases of this problem and give a conjecture for the general case that is proved to be assymptotically correct.