On the Ramsey number of the Brauer configuration. Issue 2 (14th April 2020)

• Record Type: Journal Article
• Authors: Chapman, Jonathan; Prendiville, Sean
• Is Part Of: Bulletin of the London Mathematical Society. Volume 52:Issue 2(2020)
• Page Start: 316
• Page End: 334
• Publication Date: 2020-04-14
• Subjects: 05D10 -- 11B30 (primary); Mathematics -- Periodicals; 510
• DOI: 10.1112/blms.12327
• Languages: English
• ISSNs: 0024-6093

Abstract

Abstract: We obtain a double exponential bound in Brauer's generalisation of van der Waerden's theorem, which concerns progressions with the same colour as their common difference. Such a result has been obtained independently and in much greater generality by Sanders. Using Gowers' local inverse theorem, our bound is quintuple exponential in the length of the progression. We refine this bound in the colour aspect for three‐term progressions, and combine our arguments with an insight of Lefmann to obtain analogous bounds for the Ramsey numbers of certain non‐linear quadratic equations.