Forbidden intersections for codes

Determining the maximum size of a (Formula presented.) -intersecting code in (Formula presented.) was a longstanding open problem of Frankl and Füredi, solved independently by Ahlswede and Khachatrian and by Frankl and Tokushige. We extend their result to the setting of forbidden intersections, by showing that for any (Formula presented.) and (Formula presented.) large compared with (Formula presented.) (but not necessarily (Formula presented.)) that the same bound holds for codes with the weaker property of being (Formula presented.) -avoiding, that is, having no two vectors that agree on exactly (Formula presented.) coordinates. Our proof proceeds via a junta approximation result of independent interest, which we prove via a development of our recent theory of global hypercontractivity: we show that any (Formula presented.) -avoiding code is approximately contained in a (Formula presented.) -intersecting junta (a code where membership is determined by a constant number of coordinates). In particular, when (Formula presented.), this gives an alternative proof of a recent result of Eberhard, Kahn, Narayanan and Spirkl that symmetric intersecting codes in (Formula presented.) have size (Formula presented.).