Given a sequence of positive integers p = (p_1,... ,p_n), let S_p denote the family of all sequences of positive integers x = (x_1,... ,x_n) such that X_i ≤ p_i for all i. Two families of sequences (or vectors), A, B is contained in S_p, are said to be r-cross-intersecting if no matter how we select x ∈ A and y ∈ B, there are at least r distinct indices i such that x_i = y_i. We determine the maximum value of |A| • |B| over all pairs of r-cross-intersecting families and characterize the extremal pairs for r ≥ 1, provided that min p_i ＞ r + 1. The case 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, Frankl, Fueredi, Livingston, Moon, and Tokushige, and answers a question of Zhang. The special case r = 1 has also been settled recently by Borg.
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机译：给定一个正整数序列p =（p_1，...，p_n），令S_p表示所有正整数序列x =（x_1，...，x_n）的族，使得所有i的X_i≤p_i。如果无论我们如何选择x∈A和y∈B，至少存在r个不同的索引i，则S_p中包含两个系列的序列（或向量）A，B是r交叉的。 x_i = y_i。我们确定| A |的最大值•| B |假设min p_i＞ r +1。在最小p_i≤r + 1的情况下，r≥1的所有极端对都具有特征。对于这种情况，我们有一个猜想，可以在其他假设下进行验证。我们的研究结果归纳并加强了Berge，Frankl，Fueredi，Livingston，Moon和Tokushige先前的一些研究结果，并回答了Zhang的问题。博格最近也解决了特殊情况r = 1。
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