Etude du graphe divisoriel 6
Eric Saias
TL;DR
The paper analyzes the divisor graph restricted to integers up to $x$ and quantifies how many integers can be covered by $y$ disjoint simple paths, denoted by $F(x,y)$. By leveraging the Schinzel–Szekeres sets $\\mathcal A(x)$ and $\\mathcal B(x)$, their structural links, and a decomposition into components, it introduces and bounds two auxiliary functions $R(x,z)$ and $G(x,y)$ to capture the contribution of long chains. The main result provides tight two-sided bounds $c\frac{x}{\log^+(x/y)} \le F(x,y) \le K\frac{x}{\log^+(x/y)}$ for all $x\ge 2y\ge 2$, answering Erdős's question about the growth rate when using multiple disjoint chains. The paper also proves corresponding asymptotics for $G(x,y)$ and connects these to the underlying arithmetic structure given by $\\mathcal A(x)$ and $\\mathcal B(x)$, yielding a comprehensive view of long-chain phenomena in the divisor graph with explicit, usable constants.
Abstract
The divisor graph is the non oriented graph whose vertices are the positive integers, and edges are the {a,b} such that a divides b or b divides a. Let F(x,y) be the maximum number of integers<= x belonging in one of y pairwise disjoint simple path of the restriction of the divisor graph to integers <= x. Our main result is the following. There exist two real numbers K >c>0 such that for every x and y with x>=2y>=2 , we have cx / log(x/y) <= F(x,y) <= Kx / log(x/y). It answers a question of Erdös.