Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Computation of the Expected Number of Posts in a Finite Random Graph
  Order   A random graph order is a partial order achieved by independently sprinkling
relations on a vertex set (each with probability $p$) and adding relations to
satisfy the requirement of transitivity. A \textit{post} is an element in a
partially ordered set which is related to every other element. Alon et al.\
\cite{Alon} proved a result for the average number of posts among the elements
$\{1,2,...,n\}$ in a random graph order on $\mathbb{Z}$. We refine this result
by providing an expression for the average number of posts in a random graph
order on $\{1,2,...,n\}$, thereby quantifying the edge effects associated with
the elements $\mathbb{Z}\backslash\{1,2,...,n\}$. Specifically, we prove that
the expected number of posts in a random graph order of size $n$ is
asymptotically linear in $n$ with a positive $y$-intercept. The error
associated with this approximation decreases monotonically and rapidly in $n$,
permitting accurate computation of the expected number of posts for any $n$ and
$p$. We also prove, as a lemma, a bound on the difference between the Euler
function and its partial products that may be of interest in its own right.

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The categories of this paper are: atom-ph
0
The categories of this paper are: hep-ph
25
The categories of this paper are: patt-sol
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
10
The categories of this paper are: q-bio
27
The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
35
The categories of this paper are: mtrl-th
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The categories of this paper are: q-fin
11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
21
The categories of this paper are: adap-org