Diversity of uniform intersecting families

A family F⊂2[n] is called intersecting, if any two of its sets intersect. Given an intersecting family, its diversity is the number of sets not passing through a fixed most popular element of the ground set. Peter Frankl made the following conjecture: for n>3k>0 any intersecting family F⊂[n]k has diversity at most n−3k−2. This is tight for the following “two out of three” family: {F∈[n]k:|F∩[3]|≥2}. In this note we prove this conjecture for n≥ck, where c is a constant independent of n andk. In the last section, we discuss the case 2k