Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Problem of Erd\"{o}s Concerning Lattice Cubes   This paper studies a problem of Erd\"{o}s concerning lattice cubes. Given an
$N \times N \times N$ lattice cube, we want to find the maximum number of
vertices one can select so that no eight corners of a rectangular box are
chosen simultaneously. Erd\"{o}s conjectured that it has a sharp upper bound,
which is $O(N^{11/4})$, but no example that large has been found yet. We start
approaching this question for small $N$ using the method of exhaustion, and we
find that there is not necessarily a unique maximal set of vertices (counting
all possible symmetries). Next, we study an equivalent two-dimensional version
of this problem looking for patterns that might be useful for generalizing to
the three-dimensional case. Since an $n \times n$ grid is also an $n \times n$
matrix, we rephrase and generalize the original question to: what is the
minimum number $\alpha(k,n)$ of vertices one can put in an $n \times n$ matrix
with entries 0 and 1, such that every $k \times k$ minor contains at least one
entry of 1, for $1 \leq k \leq n$? We discover some interesting formulas and
asymptotic patterns that shed new light on the question.

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The categories of this paper are: gr-qc
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: stat
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: q-alg
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The categories of this paper are: funct-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin