On symmetric intersecting families
Abstract
We make some progress on a question of Babai from the 1970s, namely: for n, k ∈ N with k n/2, what is the largest possible cardinality s(n,k) of an intersecting family of k-element subsets of \1,2,…,n\ admitting a transitive group of automorphisms? We give upper and lower bounds for s(n,k), and show in particular that s(n,k) = o (n-1k-1) as n ∞ if and only if k = n/2 - ω(n)(n/ n) for some function ω(·) that increases without bound, thereby determining the threshold at which `symmetric' intersecting families are negligibly small compared to the maximum-sized intersecting families. We also exhibit connections to some basic questions in group theory and additive number theory, and pose a number of problems.
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