On symmetric intersecting families

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 [Formula presented] as n→∞ if and only if k=n∕2−ω(n)(n∕logn) 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.