Diversity of uniform intersecting families

09/08/2017

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by   Andrey Kupavskii, et al.

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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 the most popular element of the ground set. Frankl made the following conjecture: for n> 3k>0 any intersecting family f⊂[n] k has diversity at most n-3 k-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 and k.