The extremal conjecture for intersecting families of binary sequences

Let $n$ be a positive integer, let $\vec t=(t_1,t_2)$ be a positive integer vector with $t_1+t_2\le n$, and let $\mathcal F\subset 2^{[n]}$ be $\vec t$-intersecting. For subsets $X_1,X_2\subset[n]$, define

$$\mathcal K(X_1,X_2,\vec t)=\Big\{(y_1,\ldots,y_n)\in 2^{[n]}: \big|\{j:j\in X_i\text{ and }y_j=i\}\big|\ge \frac{n_i+t_i}{2}\text{ for }i=1,2\Big\},$$

where $n_i=|X_i|$.

Extremal intersection conjecture.

$$|\mathcal F|\le \max_{X_1,X_2}|\mathcal K(X_1,X_2,\vec t)|.$$

The conjecture proposes that the largest $\vec t$-intersecting family is obtained by one of the explicitly constructed families $\mathcal K(X_1,X_2,\vec t)$. The paper establishes the corresponding asymptotic lower-bound construction for fixed positive $t_1,t_2$ when $n$ tends to infinity, but the stated extremal inequality is not resolved here.