Beyond Frankl's conjecture: Maximum sum of sizes of cross-intersecting families with restricted conditions

Let F and G be two families of subsets of $[n]$. We say that F and G are cross-intersecting if F ∩ G ≠ ∅ for all F ∈ F and G ∈ G. Let n, k₁, k₂, m, r, t be positive integers satisfying n > k₁ + k₂. Suppose that F ⊆ [n]k₁ and G ⊆ [n]k₂ are two cross-intersecting families such that ∈ [n]k₁ : |F ∩ [m]| ≥ r\ ⊆ F and F is t-intersecting. In this paper, we determine the maximum sum |F| + |G| under these assumptions. By taking k₁=k₂+t-1, m = k₂+t+s-1 and r=k₂+t-1 for s≥ 0, our result recovers a conjecture of Frankl {frankl2016new}, which was recently confirmed by Wu, Feng, and Li {wu2025proof}. Further, we establish a complete characterization of all extremal families. Our approach extends the Ahlswede--Khachatrian generating set method to handle the simultaneous constraints of cross-intersection and t-intersection, subject to additional structural conditions.