Stability for the complete intersection theorem, and the forbidden intersection problem of Erdos and Sos

A family F of sets is said to be t -intersecting if jA ∩ Bj ≥ t for any A; B ∈ F . The seminal Complete Intersection Theorem of Ahlswede and Khachatrian (1997) gives the maximal size f .n; k; t / of a t -intersecting family of k-element subsets of OEn. D 11; : : : ; no, together with a characterisation of the extremal families, solving a longstanding problem of Frankl. The forbidden intersection problem, posed by Erdos and Sos in 1971, asks for a determination of the maximal size g.n; k; t / of a family F of k-element subsets of OEn. such that jA ∩ Bj ≠ t - 1 for any A;B ∈ F . In this paper, we show that for any fixed t 2 N, if o.n/ ≤ k ≤ n/2 - o.n/, then g.n; k; t / D f .n; k; t /. In combination with prior results, this solves the problem of Erdos and Sos for any constant t , except for the ranges n/2 - o.n/ < k < n/2 C t/2 and k < 2t . One key ingredient of the proof is the following sharp 'stability' result for the Complete Intersection Theorem: if k=n is bounded away from 0 and 1=2, and F is a t -intersecting family of k-element subsets of OEn. such that jF j ≥f .n; k; t / - O. (n-d k ) /, then there exists a family G such that G is extremal for the Complete Intersection Theorem, and jF n Gj D O. (n-d k-d ) /. This proves a conjecture of Friedgut (2008).We prove the result by combining classical 'shifting' arguments with a 'bootstrapping' method based upon an isoperimetric inequality. Another key ingredient is a 'weak regularity lemma' for families of k-element subsets of OEn., where k=n is bounded away from 0 and 1. This states that any such family F is approximately contained within a 'junta' such that the restriction of F to each subcube determined by the junta is 'pseudorandom' in a certain sense.