Spread approximations for forbidden intersections problems

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.