
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 and be two -uniform families of subsets over , where , and let be the adjacency matrix of the bipartite graph whose vertices are the subsets in and , and there is an edge between and if and only if . The pair is -almost cross intersecting if every row and column of has exactly zeros. We consider -almost cross intersecting pairs that have a circulant intersection matrix , determined by a column vector with ones followed by 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 whose…
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