Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On $(\le k)$-edges, crossings, and halving lines of geometric drawings
  of $K_n$   Let $P$ be a set of points in general position in the plane. Join all pairs
of points in $P$ with straight line segments. The number of segment-crossings
in such a drawing, denoted by $\crg(P)$, is the \emph{rectilinear crossing
number} of $P$. A \emph{halving line} of $P$ is a line passing though two
points of $P$ that divides the rest of the points of $P$ in (almost) half. The
number of halving lines of $P$ is denoted by $h(P)$. Similarly, a
$k$\emph{-edge}, $0\leq k\leq n/2-1$, is a line passing through two points of
$P$ and leaving exactly $k$ points of $P$ on one side. The number of $(\le
k)$-edges of $P$ is denoted by $E_{\leq k}(P) $. Let $\rcr(n)$, $h(n)$, and
$E_{\leq k}(n) $ denote the minimum of $\crg(P)$, the maximum of $h(P)$, and
the minimum of $E_{\leq k}(P) $, respectively, over all sets $P$ of $n$ points
in general position in the plane. We show that the previously best known lower
bound on $E_{\leq k}(n)$ is tight for $k<\lceil (4n-2) /9\rceil $ and improve
it for all $k\geq \lceil (4n-2) /9 \rceil $. This in turn improves the lower
bound on $\rcr(n)$ from $0.37968\binom{n} {4}+\Theta(n^{3})$ to
{277/729}\binom{n}{4}+\Theta(n^{3})\geq 0.37997\binom{n}{4}+\Theta(n^{3})$. We
also give the exact values of $\rcr(n)$ and $h(n) $ for all $n\leq27$. Exact
values were known only for $n\leq18$ and odd $n\leq21$ for the crossing number,
and for $n\leq14$ and odd $n\leq21$ for halving lines.

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The categories of this paper are: cond-mat
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The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
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The categories of this paper are: physics
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The categories of this paper are: ao-sci
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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-th
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The categories of this paper are: comp-gas
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The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
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The categories of this paper are: q-fin
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The categories of this paper are: chao-dyn
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The categories of this paper are: bayes-an