Abstract
Given an irreducible fraction \(\frac{c}{d} \in [0,1]\), a pair \((\mathcal {A},\mathcal {B})\) is called a \(\frac{c}{d}\)-cross-intersecting pair of \(2^{[n]}\) if \(\mathcal {A}, \mathcal {B}\) are two families of subsets of [n] such that for every pair \(A \in \mathcal {A}\) and \(B\in \mathcal {B}\), \(|A \cap B|= \frac{c}{d}|B|\). Mathew et al. (Graphs Comb 37:471–484, 2019) proved that \(|\mathcal {A}||\mathcal {B}|\le 2^n\) if \((\mathcal {A}, \mathcal {B})\) is a \(\frac{c}{d}\)-cross-intersecting pair of \(2^{[n]}\) and characterized all the pairs \((\mathcal {A},\mathcal {B})\) with \(|\mathcal {A}||\mathcal {B}|=2^n\), such a pair also is called a maximal \(\frac{c}{d}\)-cross-intersecting pair of \(2^{[n]}\), when \(\frac{c}{d}\in \{0,\frac{1}{2}, 1\}\). In this note, we characterize all the maximal \(\frac{c}{d}\)-cross-intersecting pairs \((\mathcal {A},\mathcal {B})\) when \(0<\frac{c}{d}<1\) and \(\frac{c}{d}\not =\frac{1}{2}\), this result answers a question proposed by Mathew et al. (2019).
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The funding has been received from National Natural Science Foundation of China with Grant no. 12071453
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The work was supported by the National Natural Science Foundation of China (No. 12071453), the National Key R and D Program of China (2020YFA0713100), and the Innovation Program for Quantum Science and Technology, China (2021ZD0302902).
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Wang, H., Hou, X. Maximal Fractional Cross-Intersecting Families. Graphs and Combinatorics 39, 81 (2023). https://doi.org/10.1007/s00373-023-02674-4
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DOI: https://doi.org/10.1007/s00373-023-02674-4