## A Computation of the Expected Number of Posts in a Finite Random Graph Order

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Bombelli, Luca

Seggev, Itai

Watson, Sam

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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.

Comment: 11 pages, 6 figures; version 2 adds missing .bbl file for bibliography

Comment: 11 pages, 6 figures; version 2 adds missing .bbl file for bibliography

##### Keywords

Mathematics - Combinatorics, General Relativity and Quantum Cosmology, Mathematical Physics, 05C80