Forbidden intersections for codes

Peter Keevash, Noam Lifshitz, Eoin Long, Dor Minzer

Published 2021-03-08Version 1

Determining the maximum size of a $t$-intersecting code in $[m]^n$ was a longstanding open problem of Frankl and F\"uredi, 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 $m>2$ and $n$ large compared with $t$ (but not necessarily $m$) that the same bound holds for codes with the weaker property of being $(t-1)$-avoiding, i.e.\ having no two vectors that agree on exactly $t-1$ 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 $(t-1)$-avoiding code is approximately contained in a $t$-intersecting junta (a code where membership is determined by a constant number of co-ordinates). In particular, when $t=1$ this gives an alternative proof of a recent result of Eberhard, Kahn, Narayanan and Spirkl that symmetric intersecting codes in $[m]^n$ have size $o(m^n)$.