Circulant almost cross intersecting families

TL;DR

This paper studies special bipartite graphs with intersection properties, constructing and analyzing families of subsets with circulant intersection matrices, revealing new extremal configurations and bounds in combinatorial set systems.

Contribution

It introduces the concept of circulant almost cross intersecting families, providing constructions and bounds for various parameter ranges, extending understanding of intersection matrices in combinatorics.

Findings

Constructed pairs with intersection matrix $C_{p,q}$ for many parameter ranges.

Proved upper bounds matching some constructions.

Identified maximal submatrices in intersection graphs for certain parameters.

Abstract

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.