Stability analysis for k-wise intersecting families

We consider the following generalization of the seminal Erdos Ko-Rado theorem, due to Frankl [5]. 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 F-1, ..., F-k is an element of F, boolean AND(k)(i=1) F-i not equal theta .If r <= (k-1)n/k , then vertical bar F vertical bar <= ( (n-1)(r-1)) . We prove a stability version of this theorem, analogous to similar results of Dinur-Friedgut, Keevash-Mubayi and others for the EKR 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.