The Erdős--Moser sum-free set problem via improved bounds for $k$-configurations

TL;DR

This work advances the Erdős–Moser sum-free set problem by proving a substantially sharper bound for the appearance of $k$-configurations in dense subsets of abelian groups: for any $\alpha\in(0,1]$ and $k\ge2$, if $N\ge\exp(C k^{68}\log(2/\alpha)^{16})$, then every $A\subseteq [N]$ with density at least $\alpha$ contains a non-degenerate $k$-configuration. The authors achieve this through a tightly quantified graph counting lemma for binary systems of linear forms, combined with the Kelley–Meka density-increment strategy and Bohr-set technology, enabling a transition from $k$-configurations to Erdős–Moser-type sum-free subsets. As an application, they derive a lower bound for the Erdős–Moser problem of the form $|B| \ge (\log|A|)^{1+c}$ with $c\in(0,1/68)$ (explicitly $c=1/69$ is given), improving the known bounds to a Sanders-type shape. The results synthesize binary-linear-form methods with additive- combinatorics techniques to yield a sharper, more general framework for sum-free configurations in integers.

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 $N \geq \exp((k\log(2/α))^{O(1)})$, any subset $A \subseteq [N]$ of density at least $α$ contains a $k$-configuration. This improves on the previously best known bound $N \geq \exp((2/α)^{O(k^2)})$, 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 $(\log|A|)^{1+Ω(1)}$ 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 Erdős--Moser sum-free set problem of the same shape as the best known bound, established by Sanders.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['thm:graph_counting']}.
• Theorem 3.1
• proof : Proof of Theorem \ref{['thm:main_result']} assuming Theorem \ref{['thm:general_groups']}.
• Proposition 3.2
• proof : Proof of Theorem \ref{['thm:general_groups']} assuming Proposition \ref{['prop:density_increment']}.