Cross-Intersecting Families of Vectors

Given a sequence of positive integers $$p=(p_1,\dots ,p_n)$$ p = ( p 1 , ⋯ , p n ) , let $$S_p$$ S p denote the family of all sequences of positive integers $$x=(x_1,\ldots ,x_n)$$ x = ( x 1 , ... , x n ) such that $$x_i\le p_i$$ x i ≤ p i for all $$i$$ i . Two families of sequences (or vectors), $$A,B\subseteq S_p$$ A , B ⊆ S p , are said to be $$r$$ r -cross-intersecting if no matter how we select $$x\in A$$ x ∈ A and $$y\in B$$ y ∈ B , there are at least $$r$$ r distinct indices $$i$$ i such that $$x_i=y_i$$ x i = y i . We determine the maximum value of $$|A|\cdot |B|$$ | A | · | B | over all pairs of $$r$$ r -cross-intersecting families and characterize the extremal pairs for $$r\ge 1$$ r ≥ 1 , provided that $$\min p_i>r+1$$ min p i > r + 1 . The case $$\min p_i\le r+1$$ min p i ≤ r + 1 is quite different. For this case, we have a conjecture, which we can verify under additional assumptions. Our results generalize and strengthen several previous results by Berge, Borg, Frankl, Füredi, Livingston, Moon, and Tokushige, and answers a question of Zhang