On constant-multiple-free sets contained in a random set of integers

TL;DR

This paper extends the study of maximum size of rational multiple-free sets from fixed sets to random subsets of integers, providing probabilistic bounds for their maximum size.

Contribution

It generalizes previous deterministic results to random subsets, establishing asymptotic bounds for the maximum size of (b/a)-multiple-free sets within a probabilistically chosen set.

Findings

Maximum size of (b/a)-multiple-free sets in random sets is approximately (b/(b+p)) * p * n.

Asymptotic bounds hold with high probability as n approaches infinity.

Results connect combinatorial number theory with probabilistic methods.

Abstract

For a rational number $r>1$, a set $A$ of positive integers is called an $r$-multiple-free set if $A$ does not contain any solution of the equation $rx = y$. The extremal problem on estimating the maximum possible size of $r$-multiple-free sets contained in $[n]:={1,2,...,n}$ has been studied for its own interest in combinatorial number theory and application to coding theory. Let $a$, $b$ be positive integers such that $a<b$ and the greatest common divisor of $a$ and $b$ is 1. Wakeham and Wood showed that the maximum size of $(b/a)$-multiple-free sets contained in $[n]$ is $\frac{b}{b+1}n+O(\log n)$. In this paper we generalize this result as follows. For a real number $p\in (0,1)$, let $[n]_p$ be a set of integers obtained by choosing each element $i\in [n]$ randomly and independently with probability $p$. We show that the maximum possible size of $(b/a)$-multiple-free sets contained in $[n]_p$ is $\frac{b}{b+p}pn+O(\sqrt{pn}\log n \log \log n)$ with probability that goes to 1 as $n\to \infty$.