Best possible bounds on the number of distinct differences in intersecting families

06/09/2021

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by   Peter Frankl, et al.

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For a family ℱ, let 𝒟(ℱ) stand for the family of all sets that can be expressed as F∖ G, where F,G∈ℱ. A family ℱ is intersecting if any two sets from the family have non-empty intersection. In this paper, we study the following question: what is the maximum of |𝒟(ℱ)| for an intersecting family of k-element sets? Frankl conjectured that the maximum is attained when ℱ is the family of all sets containing a fixed element. We show that this holds if n>50klog k and k>50. At the same time, we provide a counterexample for n< 4k.