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 $n, kin mathbb{N}$ 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 [expleft(-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.
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