Erdos-Ko-Rado from intersecting shadows

A set system is called t-intersecting if every two members meet each other in at least t elements. Katona determined the minimum ratio of the shadow and the size of such families and showed that the Erdos-Ko-Rado theorem immediately follows from this result. The aim of this note is to reproduce the proof to obtain a slight improvement in the Kneser graph. We also give a brief overview of corresponding results

Intersection problems in the q-ary cube

We propose new intersection problems in the q-ary n-dimensional hypercube. The answers to the problems include the Katona's t-intersection theorem and the Erdos-Ko-Rado theorem as special cases. We solve some of the basic cases of our problems, and for example we get an Erdos-Ko-Rado type result for t-intersecting k-uniform families of multisets with bounded repetitions. Another case is obtained by counting the number of lattice points in a polytope having an intersection property. © 2016 Elsevier Inc