论文标题
最大的分数交叉截断家庭
Maximal fractional cross-intersecting families
论文作者
论文摘要
给定一个不可约的分数$ \ frac {c} {d} \ in [0,1] $，一对$（\ Mathcal {a}，\ Mathcal {b}）$称为$ \ frac {c} {c} {d} {d} {d} {d} { \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，Ray和Srivastava [{\ it \小部分交叉相交的家庭，图形和梳子，2019}]证明了$ | \ Mathcal {a} || \ Mathcal {b} $ \ frac {c} {d} $ - $ 2^{[n]} $的交叉交流对，并表征所有对$（\ Mathcal {a}，\ Mathcal {a}，\ Mathcal {b}）$，带有$ | \ | \ | \ Mathcal {a} $ 2^{[n]} $的CD $ -CROSS-CROSSINTECTING对，当时$ \ frac cd \ in \ {0，\ frac12，1 \} $。在本说明中，我们表征了所有最大$ \ frac CD $ -CROSS与交叉交叉$（\ Mathcal {a}，\ Mathcal {b}）$当$ 0 <\ frac {c} {c} {c} {d} {d} {d} <1 $ and $ \ frac cd cd \ frac cd cd \ frac cd \ frac cd \ frac 12 $，the Mathiv and ass and ass and ass and ass and ass and ass and ass and ass and ass and ass and。
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, Ray, and Srivastava [{\it\small Fractional cross intersecting families, Graphs and Comb., 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 $\frac cd$-cross-intersecting pair of $2^{[n]}$, when $\frac cd\in\{0,\frac12, 1\}$. In this note, we characterize all the maximal $\frac cd$-cross-intersecting pairs $(\mathcal{A},\mathcal{B})$ when $0<\frac{c}{d}<1$ and $\frac cd\not=\frac 12$, this result answers a question proposed by Mathew, Ray, and Srivastava (2019).
