An Overlap Construction for Relative Linear Extension Ratios

Chan and Pak introduced the relative linear extension ratio ρ(P, x) = e(P)/e(P − x), where e(P) is the number of linear extensions of a finite poset P, and let ν(c, d) be the least number of elements of a poset that realizes ρ(P, x) = d/c. They proved that ν(c, d) ≤ d/c + O(log d log log d) for d ≥ 3c, and asked whether the hypothesis d ≥ 3c can be relaxed to d ≥ (1 + ε)c or removed. We prove the fixed-gap form of this question: for every fixed ε > 0, ν(c, d) ≤ d/c + Oε(log d log log d) whenever d ≥ (1 + ε)c, and the implied constant is absolute once d ≥ 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 |R| = |P| + |Q| − 1 and an element z such that ρ(R, z) = ρ(P, x) + ρ(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/c < 2, with h = d − c, the size bound the construction can certify is at least ⌊c/h⌋, so the method reaches the stated error term only when h is at least of order c/(log c log log c). The remaining obstruction to removing the hypothesis is a short-interval problem for sums of partial quotients. The deductive part of the argument has been checked with the Lean proof assistant.This is a preprint and has not been peer reviewed. The record includes the paper, LaTeX source, and ancillary formal-verification files.