  We consider the following generalization of the seminal Erd\H{o}s-Ko-Rado
theorem, due to Frankl. For some k>=2, let F be a k-wise intersecting family of
r-subsets of an n element set X, i.e. for any k sets F1,...,Fk in F, their
intersection is nonempty. If r <= ((k-1)n)/k, then |F|<= {n-1 \choose r-1}. We
prove a stability version of this theorem, analogous to similar results of
Dinur-Friedgut, Keevash-Mubayi and others for the Erd\H{o}s-Ko-Rado theorem.
The technique we use is a generalization of Katona's circle method, initially
employed by Keevash, which uses expansion properties of a particular Cayley
graph of the symmetric group.
