Forbidden subgraphs in divisor graphs and an Erd\H{o}s divisibility problem

TL;DR

This paper determines the asymptotic maximum size and number of subsets of {1,...,n} avoiding certain divisibility patterns, using graph theory and local statistics of divisor graphs.

Contribution

It establishes effective formulas for the maximum size and count of divisibility-avoiding subsets, generalizing to forbidden subgraphs in divisor graphs.

Findings

Maximum size of such subsets is approximately c_2 * n.

Number of such subsets grows like beta_2^n.

General result applies to any finite family of forbidden divisor subgraphs.

Abstract

Erd\H{o}s asked for the largest size $f(n)$ of a subset of $\{1,\dots,n\}$ with no element dividing two others. We show that $f(n)=c_2\,n+o(n)$ for an effectively computable constant $c_2$, and moreover that the number $q(n)$ of such subsets satisfies $q(n)=\beta_2^{n+o(n)}$ for a computable constant $\beta_2$. To prove this, we recast the divisibility constraint as forbidding a certain directed subgraph in the divisor graph on $\{1,\dots,n\}$ and prove a more general result: for any finite family of connected forbidden subgraphs of the divisor graph, both the extremal density and counting rate are effectively computable. The proof uses a theorem of McNew on local statistics of divisor graphs.