Etude du graphe divisoriel 6

TL;DR

This paper investigates the divisor graph of positive integers, establishing bounds on the maximum number of integers within certain disjoint paths, thus answering a question posed by Erdős.

Contribution

It provides new bounds on the function F(x,y), relating to disjoint paths in the divisor graph, with explicit constants, addressing Erdős's question.

Findings

Existence of constants c and K such that cx / log(x/y) <= F(x,y) <= Kx / log(x/y)

F(x,y) scales approximately with x / log(x/y)

Answers a long-standing question of Erdős about divisor graphs

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 Erd\"os.