2025/01/17 by Adrian Beker, Beker, Adrian
Engineering · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Point processes and geometric inequalities #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2501.10203
openalex publication_date 2025/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A k-configuration is a collection of k distinct integers x1,…,xk together with their pairwise arithmetic means (xi+xj)/(2) for 1 ≤ i < j ≤ 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 ≥ exp((klog(2/α))O(1)), any subset A ⊆ [N] of density at least α contains a k-configuration. This improves on the previously best known bound N ≥ exp((2/α)O(k2)), due to Shao. As a consequence, it follows that any finite non-empty set A ⊆ ℤ contains a subset B ⊆ A of size at least (log|A|)1+Ω(1) such that b1+b2 \not∈ A for any distinct b1,b2 ∈ 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.