# On a conjecture of Erd\H{o}s about sets without $k$ pairwise coprime integers

## TL;DR

This paper proves a conjecture by Erdős regarding the maximum size of sets without $k+1$ pairwise coprime integers, confirming a specific structure of such sets for large enough $n$.

## Contribution

The paper resolves an open problem posed by Chen and Zhou, confirming Erdős's conjecture and providing new insights into the structure of sets avoiding $k+1$ pairwise coprime integers.

## Key findings

- Confirmed Erdős's conjecture for all sufficiently large n.
- Characterized the structure of extremal sets avoiding $k+1$ coprime integers.
- Extended previous results by Chen and Zhou on the conjecture.

## Abstract

Let $\mathbb{Z}^{+}$ be the set of positive integers. Let $C_{k}$ denote all subsets of $\mathbb{Z}^{+}$ such that neither of them contains $k + 1$ pairwise coprime integers and $C_k(n)=C_k\cap \{1,2,\ldots,n\}$. Let $f(n, k) = \text{max}_{A \in C_{k}(n)}|A|$, where $|A|$ denotes the number of elements of the set $A$. Let $E_k(n)$ be the set of positive integers not exceeding $n$ which are divisible by at least one of the primes $p_{1}, \dots{}, p_{k}$, where $p_{i}$ denote the $i$th prime number. In 1962, Erd\H{o}s conjectured that $f(n, k) = |E(n,k)|$ for every $n \ge p_{k}$. Recently Chen and Zhou proved some results about this conjecture. In this paper we solve an open problem of Chen and Zhou and prove several related results about the conjecture.