# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1907.11632