On symmetric intersecting families. (May 2020) - Ellis, David; Kalai, Gil; Narayanan, Bhargav
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 − 1 k − 1 ) as n → ∞ if and only if k = n ∕ 2 − ω ( n ) ( n ∕ log 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.