Stability Analysis for k-wise Intersecting Families

We consider the following generalization of the seminal Erdős–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 F1, . . . , Fk ∈ F , ∩ k i=1Fi 6= ∅. If r ≤ (k − 1)n k , then |F| ≤ ( 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.