An Overlap Construction for Relative Linear Extension Ratios

Chan and Pak introduced the relative linear extension ratio $\rho (P,x)=e(P)\mathrm{/}e(P-x)$, where $e(P)$ is the number of linear extensions of a finite poset $P$, and let $\nu (c,d)$ be the least number of elements of a poset that realizes $\rho (P,x)=d\mathrm{/}c$. They proved that $\nu (c,d)\le d\mathrm{/}c+O(\log d\log \log d)$ for $d\ge 3c$, and asked whether the hypothesis $d\ge 3c$ can be relaxed to $d\ge (1+\epsilon )c$ or removed. We prove the fixed-gap form of this question: for every fixed $\epsilon >0$, $\nu (c,d)\le \frac{d}{c}+O_{\epsilon }(\log d\log \log d)$ whenever $d\ge (1+\epsilon )c$, and the implied constant is absolute once $d\ge 2c$. The new ingredient is a one-element overlap construction: if $x$ is minimal in $P$ and $y$ is minimal in $Q$, then there is a poset $R$ with $\mathrm{\mid }R\mathrm{\mid }=\mathrm{\mid }P\mathrm{\mid }+\mathrm{\mid }Q\mathrm{\mid }-1$ and an element $z$ such that $\rho (R,z)=\rho (P,x)+\rho (Q,y)-1$. Together with the continued-fraction construction of Chan and Pak and Rukavishnikova’s tail bound for sums of partial quotients, this removes the factor $3$ in their range. We also show that the fixed-gap hypothesis is essentially optimal for this construction. In the range $1<d\mathrm{/}c<2$, with $h=d-c$, the size bound the construction can certify is at least $\lfloor c\mathrm{/}h\rfloor$, so the method reaches the stated error term only when $h$ is at least of order $c\mathrm{/}(\log c\log \log c)$. The remaining obstruction to removing the hypothesis is a short-interval problem for sums of partial quotients, which we describe. The deductive part of the argument has been checked with the Lean proof assistant.