CATALOGUE 159
Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds
This catalogue concerns density bounds for sets avoiding arithmetic progressions of a fixed length, strong enough to force convergence of their reciprocal sums. The manuscript constructs density increments on triangular cells retaining earlier polynomial constraints, then controls dimension and width losses to close a finite iteration.
1. Papers in official catalogue order
These summaries give the principal scope; consult the cited original results for the complete hypotheses and statements.
Page counts include references. On narrow screens, scroll the table sideways.
| # | Manuscript and overview | Principal claim | Role and available overview |
|---|---|---|---|
| 01 | Quasipolynomial Bounds for Arithmetic Progressions →Short name: Quasipolynomial Bounds for Arithmetic ProgressionsSeptember 23, 2026 · 198 pagesOriginal paper (GitHub) ↗ | For fixed $k\ge3$, bounds the largest nonconstant-k-term-progression-free subset by $C_kN\exp\{-c_k(\log N)^{\varepsilon_k}\}$. Every set of positive integers with divergent reciprocal sum contains progressions of every finite length, nonconstant for lengths at least two. Theorem 1.1; Corollary 1.2 · p. 4 ↗ | Controls triangular-cell dimensions and widths from higher weights, iterates a fixed multiplicative density gain, and passes from the finite bound to reciprocal sums by dyadic summation. Read the proof overview → |
2. Sources and checking scope
[OpenAI26Catalog] denotes OpenAI’s OpenAI Research Catalog. The official subject is Combinatorics, and the official catalogue number is 159. We use the OpenAI Research Catalog at commit 1581d4cbdd69a0faf34b2089614fe2a814316a6a, with the catalogue entry on overview p. 18 and the manuscript dated September 23, 2026.
[OpenAI26Catalog, Catalogue 159, p. 18]
The article on Quasipolynomial Bounds for Arithmetic Progressions checks the principal hypotheses and quantifiers, triangular iteration, inversion of the threshold, summation for reciprocal sums, and statements and application points of the direct external inputs. It does not independently verify the entire analytic construction. Unchecked technical portions are identified in the article. Proof connections to other catalogues remain uninvestigated; this does not mean that no such dependencies exist.