We make some progress on a question of Babai from the 1970s, namely: for n,kβN with kβ€n/2, what is the largest possible cardinality
s(n,k) of an intersecting family of k-element subsets of {1,2,β¦,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((kβ1nβ1β)) as nββ if and only if k=n/2βΟ(n)(n/logn) for some function
Ο(β ) 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.Comment: Minor change to the statement (and proof) of Theorem 1.4; the authors
thank Nathan Keller and Omri Marcus for pointing out a mistake in the
previous versio