A new Bollobás-type inequality and applications to t-intersecting families of sets

Bollobásʹs theorem is a fundamental result in the theory of extremal set systems. A related result due to Lovász gives bounds on the size of the kernel of an intersecting family of sets. We give a new generalization of Bollobasʹs theorem. Applications of this result are given and in particular we obtain bounds on the size of the kernel of a t-intersecting family of sets, generalizing Lovaszʹs result