Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On maximal isolation sets in the uniform intersection matrix   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.

36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
30
The categories of this paper are: q-alg
18
The categories of this paper are: econ
20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
25
The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
24
The categories of this paper are: dg-ga