On a Ramsey–Turán variant of Roth's theorem
Abstract A classical theorem of Roth states that the maximum size of a solution‐free set of a homogeneous linear equation in is if and only if the sum of the coefficients of is 0. In this paper, we prove a Ramsey–Turán variant of Roth's theorem, with respect to a natural notion of “structured” sets introduced by Erdős and Sárközy in the 1970s. Namely, we show that the following statements are equivalent: Every solution‐free set of in with has size . There exists a non‐empty subset of coefficients of with zero sum.
Authors
- Matija Bucić (ORCID: https://orcid.org/0000-0002-1055-3309)
- Micha Christoph
- Varun Sivashankar
- Jaehoon Kim (ORCID: https://orcid.org/0000-0002-1411-4186)
- Hyunwoo Lee
Institutions
- University of Vienna (AT)
- Korea Advanced Institute of Science and Technology (KR)
- Princeton University (US)
- ETH Zurich (CH)
- National Institute for Mathematical Sciences (KR)
Publication Details
- Journal
- Bulletin of the London Mathematical Society
- Published
- 2026-07-24
- DOI
- https://doi.org/10.1112/blms.70436
- Primary Topic
- Limits and Structures in Graph Theory
- Type
- article
- Field-Weighted Citation Impact
- 0.00
Funders
- National Science Foundation
- Princeton University
- Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung
- National Research Foundation of Korea
- Institute for Basic Science