On symmetric intersecting families

A family of sets is said to be \emph{symmetric} if its automorphism group is transitive, and \emph{intersecting} if any two sets in the family have nonempty intersection. Our purpose here is to study the following question: for $n, k\in \mathbb{N}$ with $k \le n/2$, how large can a symmetric intersecting family of $k$-element subsets of $\{1,2,\ldots,n\}$ be? As a first step towards a complete answer, we prove that such a family has size at most \[\exp\left(-\frac{c(n-2k)\log n}{k( \log n - \log k)} \right) \binom{n}{k},\ ] where $c > 0$ is a universal constant. We also describe various combinatorial and algebraic approaches to constructing such families.