On maximal isolation sets in the uniform intersection matrix

TL;DR

This paper investigates the maximal isolation sets in the uniform intersection matrix, providing optimal constructions and bounds, and explores related triangular submatrix structures with tight bounds based on combinatorial theorems.

Contribution

It establishes the maximum size of identity submatrices and isolation sets in the uniform intersection matrix, offering new constructions and proving their optimality for large parameters.

Findings

Largest identity submatrix size is k-2t+2.

Constructed large isolation sets matching upper bounds.

Provided tight bounds for maximal triangular isolation submatrices.

Abstract

Let $A_{k,t}$ be the matrix that represents the adjacency matrix of the intersection bipartite graph of all subsets of size $t$ of $\{1,2,...,k\}$. We give constructions of large isolation sets in $A_{k,t}$, where, for a large enough $k$, our constructions are the best possible. We first prove that the largest identity submatrix in $A_{k,t}$ is of size $k-2t+2$. Then we provide constructions of isolations sets in $A_{k,t}$ for any $t\geq 2$, as follows: \begin{itemize} \item If $k = 2t+r$ and $0 \leq r \leq 2t-3$, there exists an isolation set of size $2r+3 = 2k-4t+3$. \item If $k \geq 4t-3$, there exists an isolation set of size $k$. \end{itemize} The construction is maximal for $k\geq 4t-3$, since the Boolean rank of $A_{k,t}$ is $k$ in this case. As we prove, the construction is maximal also for $k = 2t, 2t+1$. Finally, we consider the problem of the maximal triangular isolation submatrix of $A_{k,t}$ that has ones in every entry on the main diagonal and below it, and zeros elsewhere. We give an optimal construction of such a submatrix of size $({2t \choose t}-1) \times ({2t \choose t}-1)$, for any $t \geq 1$ and a large enough $k$. This construction is tight, as there is a matching upper bound, which can be derived from a theorem of Frankl about skew matrices.