  A family $F$ of sets is said to be $t$-intersecting if $|A \cap B| \geq t$
for any $A,B \in F$. 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 $[n]=\{1,2,\ldots,n\}$, together with a
characterisation of the extremal families.
  The forbidden intersection problem, posed by Erd\H{o}s and S\'{o}s in 1971,
asks for a determination of the maximal size $g(n,k,t)$ of a family $F$ of
$k$-element subsets of $[n]$ such that $|A \cap B| \neq t-1$ for any $A,B \in
F$.
  In this paper, we show that for any fixed $t \in \mathbb{N}$, if $o(n) \leq k
\leq n/2-o(n)$, then $g(n,k,t)=f(n,k,t)$. In combination with prior results,
this solves the above problem of Erd\H{o}s and S\'{o}s for any constant $t$,
except for in the ranges $n/2-o(n) < k < n/2+t/2$ 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 $[n]$ such that
$|F| \geq f(n,k,t) - O(\binom{n-d}{k})$, then there exists a family $G$ such
that $G$ is extremal for the Complete Intersection Theorem, and $|F \setminus
G| = O(\binom{n-d}{k-d})$. We believe this result to be of interest in its own
right; indeed, it proves a conjecture of Friedgut from 2008. We prove it 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 $[n]$, 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.
