The Erdős–Ko–Rado-type conjecture for L-intersecting families

Let $0<s\leq k\leq n$ be positive integers. Let $L=\{\ell_1,\ldots,\ell_s\}$ be a set of $s$ positive integers such that $0<\ell_1<\cdots<\ell_s$. Suppose that $n>k^2-k+1$. Let $\mathcal F=\{F_1,\ldots,F_m\}$ be an $L$-intersecting, $k$-uniform family of subsets of $[n]$, meaning that $|F_i\cap F_j|\in L$ for every distinct $F_i,F_j\in\mathcal F$. The Erdős–Ko–Rado-type conjecture for $L$-intersecting families. Then

$$m\leq {n-\ell_1\choose s}.$$

Moreover, if

$$|\mathcal F|={n-\ell_1\choose s},$$

then there exists a set $T\in {[n]\choose \ell_1}$ such that $T\subseteq F$ for every $F\in\mathcal F$. This conjecture extends the preceding bound and its equality characterization from the proven range to the stated condition $n>k^2-k+1$; the projective-plane example motivating it shows why the threshold is natural. Its resolution is not indicated in the supplied text.