Non-empty pairwise cross-intersecting families

Two families A and B are cross-intersecting if A boolean AND B not equal & empty; for any A is an element of A and B is an element of B. We call t families A(1), A(2), & mldr;, A(t ) pairwise cross-intersecting families if A(i) and A(j) are cross-intersecting for 1 <= i < j <= t. Additionally, if A(i) not equal & empty; for each j is an element of[t], then we say that A(1), A(2), & mldr;, A(t )are non-empty pairwise cross-intersecting. Let A(1) subset of (([n])(k1)), A(2) subset of ([([n])(k2)), & mldr;, A(t) subset of (([n])(kt)) be non-empty pairwise cross-intersecting families with t >= 2, k(1) >= k(2) >= center dot center dot center dot >= k(t), n >= k(1)+k(2) and d(1), d(2), & mldr;, d(t) be positive numbers. In this paper, we give a sharp upper bound of & sum;(t )(j=1)d(j)|A(j)| and characterize the families A(1), A(2), & mldr;, A(t) attaining the upper bound. Our results unifies results of Frankl and Tokushige (1992) [5], Shi, Frankl and Qian (2022) [15], Huang, Peng and Wang [10], and Zhang and Feng [16]. Furthermore, our result can be applied in the treatment for some n < k(1)+k(2) while all previous known results do not have such an application. In the proof, a result of Kruskal and Katona is applied to allow us to consider only families Ai whose elements are the first |A(i)| elements in lexicographic order. We bound & sum;(t )(i=1)d(i)|A(i)| by a single variable function f(i)(R), where R is the last element of A(i) in lexicographic order, and verify that -f(i)(R) has unimodality which is stronger than the extremal result. We think that the unimodality of functions in this paper is interesting in its own, in addition to the extremal result. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.