Stability for the Complete Intersection Theorem, and the Forbidden Intersection Problem of Erd\H{o}s and S\'os

Understand

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 $g(n,k,t)=f(n,k,t)$.