Ramsey functions for quasi-progressions with large diameter
نویسندگان
چکیده
Several renowned open conjectures in combinatorics and number theory involve arithmetic progressions. Van der Waerden famously proved in 1927 that for each positive integer k there exists a least positive integer w(k) such that any 2-coloring of 1, . . . , w(k) produces a monochromatic k-term arithmetic progression. The best known upper bound for w(k) is due to Gowers and is quite large. Ron Graham [2] conjectures w(k) ≤ 2k2 , for all k. The
منابع مشابه
Ramsey Functions for Generalized Progressions
Given positive integers m and k, a k-term semi-progression of scope m is a sequence x1, x2, ..., xk such that xj+1 − xj ∈ {d, 2d, . . . ,md}, 1 ≤ j ≤ k − 1, for some positive integer d. Thus an arithmetic progression is a semi-progression of scope 1. Let Sm(k) denote the least integer for which every 2-coloring of {1, 2, ..., Sm(k)} yields a monochromatic k-term semi-progression of scope m. We ...
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Preface These are the notes based on the course on Ramsey Theory taught at Univer-sität Hamburg in Summer 2011. The lecture was based on the textbook " Ramsey theory " of Graham, Rothschild, and Spencer [44]. In fact, large part of the material is taken from that book. iii Contents Preface iii Chapter 1. Introduction 1 1.1. A few cornerstones in Ramsey theory 1 1.2. A unifying framework 4 1.3. ...
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