On Sequences Containing at Most 4 Pairwise Coprime Integers

TL;DR

This paper proves Erdős's conjecture for the case k=4, establishing that the maximum size of a set avoiding 5 pairwise coprime integers matches a specific prime-divisible set for sufficiently large n.

Contribution

The paper confirms Erdős's conjecture for k=4, extending previous results and providing a proof for large n, along with posing open problems for future research.

Findings

Confirmed the conjecture for k=4 and n≥49.

Established the equality |A(n,4)|=|E(n,4)| under given conditions.

Provided a framework for analyzing sets avoiding multiple coprime integers.

Abstract

Let $f(n,k)$ be the largest number of positive integers not exceeding $n$ from which one cannot select $k+1$ pairwise coprime integers, and let $E(n,k)$ be the set of positive integers which do not exceed $n$ and can be divided by at least one of $p_1, p_2,..., p_k$, where $p_i$ is the $i$-th prime. In 1962, P. Erd\H os conjectured that $f(n,k)=|E(n,k)|$ for all $n\ge p_k$. In 1973, S. L. G. Choi proved that the conjecture is true for $k=3$. In 1994, Ahlswede and Kachatrian disproved the conjecture for $k=212$. In this paper we prove that, for $n\ge 49$, if A(n,4) is a set of positive integers not exceeding $n$ from which one cannot select 5 pairwise coprime integers and $|A(n,4)|\ge |E(n,4)|$, then $A(n,4)=E(n,4)$. In particular, the conjecture is true for k=4. Several open problems and conjectures are posed for further research.