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 $\mathit{\text{post}}$ 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 $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ 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 $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$, thereby quantifying the edge effects associated with the elements $\mathbb{Z}\mathrm{\backslash }\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$. 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.