Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds

Proves Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. Quantitatively, for each fixed k ≥ 3, every subset of \{1,\ldots,N\} with no nonconstant k-term progression has size at most C_kN\exp[-c_k(\log N)^{\varepsilon_k}], with positive constants depending only on k. (Lean)

Manuscript

Quasipolynomial Bounds for Arithmetic Progressions

September 23, 2026 · source · cite as \cite{OAI:Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026}

We prove Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. More quantitatively, for every fixed k ≥ 3, we show

\displaystyle r_k(N)\le C_kN\exp\bigl(-c_k(\log N)^{\varepsilon_k}\bigr)

with C_k,c_k,\varepsilon_k\gt 0, where r_k(N) is the largest size of a subset of \{1,\ldots,N\} with no nonconstant k-term arithmetic progression.

Result 159 · Combinatorics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Agora members.