  We develop a new approach to approximate families of sets, complementing the
existing `$\Delta$-system method' and `junta approximations method'. The
approach, which we refer to as `spread approximations method', is based on the
notion of $r$-spread families and builds on the recent breakthrough result of
Alweiss, Lovett, Wu and Zhang for the Erd\H os--Rado `Sunflower Conjecture'.
Our approach can work in a variety of sparse settings.
  To demonstrate the versatility and strength of the approach, we present
several of its applications to forbidden intersection problems, including
bounds on the size of regular intersecting families, the resolution of the
Erd\H os--S\'os problem for sets in a new range and, most notably, the
resolution of the $t$-intersection and Erd\H os--S\'os problems for
permutations in a new range. Specifically, we show that any collection of
permutations of an $n$-element set with no two permutations intersecting in at
most (exactly) $t-1$ elements has size at most $(n-t)!$, provided $t\le
n^{1-\epsilon}$ ($t \le n^{\frac{1}{3}-\epsilon}$) for an arbitrary
$\epsilon>0$ and $n>n_0(\epsilon)$. Previous results for these problems only
dealt with the case of fixed $t$. The proof follows the structure vs.
randomness philosophy, which proved to be very efficient in proving results
throughout mathematics and computer science.
