The Ahlswede–Khachatrian conjecture for cross-t-intersecting families

Let $C(n,k,l,t)$ and $N(n,k,l,t)$ be as above. For integers $i,j$ with $0\le i+j\le n-t$, define

$$\mathcal G_{ij}(n,k,t)=\left\{F\in\binom{[n]}{k}:|F\cap[t+i+j]|\ge t+i\right\}.$$

The cross-intersecting Ahlswede–Khachatrian conjecture. One has

$$N(n,k,l,t)=\max_{0\le i+j\le n-t}|\mathcal G_{ij}(n,k,t)|\,|\mathcal G_{ji}(n,l,t)|.$$

Moreover, up to a permutation of $[n]$, an optimal cross-$t$-intersecting pair is $(\mathcal G_{ij}(n,k,t),\mathcal G_{ji}(n,l,t))$ for some $i,j$. This generalizes the Ahlswede–Khachatrian theorem for ordinary $t$-intersecting families; the corresponding cross-family classification is presented as open.