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arXiv 2005.07907math.COcs.DM
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循环几乎交叉相交族

Circulant almost cross intersecting families

Michal Parnas

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摘要：

设$\mathcal{F}$和$\mathcal{G}$是$[k] = \{1,2,...,k\}$上的两个$t$ -一致子集族，其中$|\mathcal{F}| = |\mathcal{G}|$，$C$是以$\mathcal{F}$和$\mathcal{G}$中的子集为顶点的二部图的邻接矩阵，$A\in \mathcal{F}$和$B \in \mathcal{G}$之间存在边当且仅当$A \cap B \neq \emptyset$ .对$(\mathcal{F},\mathcal{G})$是$q$ -几乎交叉相交，如果$C$的每一行和每一列都恰好有$q$零。我们考虑$q$ -几乎交叉相交对，具有循环相交矩阵$C_{p,q}$，由$p > 0$ 1后跟$q > 0$ 0的列向量确定。这个矩阵族的一端是单位矩阵，另一端是二部冠图的邻接矩阵。我们给的结构对$(\mathcal{F},\mathcal{G})$的交集矩阵是$C_{p,q}$，参数$p$和$q$的值范围很广，并在某些情况下也证明了匹配上界。具体来说，我们证明了以下参数值的结果：（1）$1 \leq p \leq 2t-1$和$1 \leq q \leq k-2t+1$。(2)$2t \leq p \leq t^2$和任何$q> 0$，其中$k \geq p+q$ . (3)对于足够大的$k$，$p$在$t$中是指数的。使用第一个结果，我们表明，如果$k \geq 4t-3$然后$C_{2t-1,k-2t+1}$是一个最大的隔离子矩阵的大小$k\times k$在$0,1$矩阵$A_{k,t}$，其行和列被标记的所有子集的大小$t$的$[k]$，有一个在条目上的行$x$和列$y$当且仅当$x,y$子集相交。

英文摘要：

Let $\mathcal{F}$ and $\mathcal{G}$ be two $t$-uniform families of subsets over $[k] = \{1,2,...,k\}$, where $|\mathcal{F}| = |\mathcal{G}|$, and let $C$ be the adjacency matrix of the bipartite graph whose vertices are the subsets in $\mathcal{F}$ and $\mathcal{G}$, and there is an edge between $A\in \mathcal{F}$ and $B \in \mathcal{G}$ if and only if $A \cap B \neq \emptyset$. The pair $(\mathcal{F},\mathcal{G})$ is $q$-almost cross intersecting if every row and column of $C$ has exactly $q$ zeros. We consider $q$-almost cross intersecting pairs that have a circulant intersection matrix $C_{p,q}$, determined by a column vector with $p > 0$ ones followed by $q > 0$ zeros. This family of matrices includes the identity matrix in one extreme, and the adjacency matrix of the bipartite crown graph in the other extreme. We give constructions of pairs $(\mathcal{F},\mathcal{G})$ whose intersection matrix is $C_{p,q}$, for a wide range of values of the parameters $p$ and $q$, and in some cases also prove matching upper bounds. Specifically, we prove results for the following values of the parameters: (1) $1 \leq p \leq 2t-1$ and $1 \leq q \leq k-2t+1$. (2) $2t \leq p \leq t^2$ and any $q> 0$, where $k \geq p+q$. (3) $p$ that is exponential in $t$, for large enough $k$. Using the first result we show that if $k \geq 4t-3$ then $C_{2t-1,k-2t+1}$ is a maximal isolation submatrix of size $k\times k$ in the $0,1$-matrix $A_{k,t}$, whose rows and columns are labeled by all subsets of size $t$ of $[k]$, and there is a one in the entry on row $x$ and column $y$ if and only if subsets $x,y$ intersect.

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