  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 $o(m^n)$.
