The Erd\H{o}s--Moser sum-free set problem via improved bounds for $k$-configurations

TL;DR

This paper improves bounds on the existence of specific configurations within dense sets, leading to new insights into sum-free subsets and advancing the Erdős–Moser problem using recent developments in additive combinatorics.

Contribution

It introduces improved bounds for $k$-configurations in dense sets, extending previous results and providing a new proof related to the Erdős–Moser sum-free set problem.

Findings

Enhanced bounds for $k$-configurations in dense sets.

New lower bounds for sum-free subsets in finite sets.

Connection to Erdős–Moser sum-free problem with improved estimates.

Abstract

A $k$-configuration is a collection of $k$ distinct integers $x_1,\ldots,x_k$ together with their pairwise arithmetic means $\frac{x_i+x_j}{2}$ for $1 \leq i < j \leq k$. Building on recent work of Filmus, Hatami, Hosseini and Kelman on binary systems of linear forms and of Kelley and Meka on Roth's theorem on arithmetic progressions, we show that, for $N \geq \exp((k\log(2/\alpha))^{O(1)})$, any subset $A \subseteq [N]$ of density at least $\alpha$ contains a $k$-configuration. This improves on the previously best known bound $N \geq \exp((2/\alpha)^{O(k^2)})$, due to Shao. As a consequence, it follows that any finite non-empty set $A \subseteq \mathbb{Z}$ contains a subset $B \subseteq A$ of size at least $(\log|A|)^{1+\Omega(1)}$ such that $b_1+b_2 \not\in A$ for any distinct $b_1,b_2 \in B$. This provides a new proof of a lower bound for the Erd\H{o}s--Moser sum-free set problem of the same shape as the best known bound, established by Sanders.