Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Pentagonal geometries with block sizes 3, 4 and 5   A pentagonal geometry PENT($k$, $r$) is a partial linear space, where every
line, or block, is incident with $k$ points, every point is incident with $r$
lines, and for each point $x$, there is a line incident with precisely those
points that are not collinear with $x$. An opposite line pair in a pentagonal
geometry consists of two parallel lines such that each point on one of the
lines is not collinear with precisely those points on the other line.
  We give a direct construction for an infinite sequence of pentagonal
geometries with block size 3 and connected deficiency graphs. Also we present
39 new pentagonal geometries with block size 4 and five with block size 5, all
with connected deficiency graphs. Consequentially we determine the existence
spectrum up to a few possible exceptions for PENT(4, $r$) that do not contain
opposite line pairs and for PENT(4, $r$) with one opposite line pair. More
generally, given $j$ we show that there exists a PENT(4, $r$) with $j$ opposite
line pairs for all sufficiently large admissible $r$. Using some new group
divisible designs with block size 5 (including types $2^{35}$, $2^{71}$ and
$10^{23}$) we significantly extend the known existence spectrum for PENT(5,
$r$).

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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: patt-sol
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The categories of this paper are: math-ph
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: supr-con
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The categories of this paper are: nlin
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The categories of this paper are: chao-dyn
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The categories of this paper are: solv-int
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The categories of this paper are: adap-org
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The categories of this paper are: chem-ph