On the Degree of Regularity of Generalized van der Waerden Triples

Let a and b be positive integers such that a ≤ b and (a, b) \not= (1, 1). We prove that there exists a 6-coloring of the positive integers that does not contain a monochromatic (a, b)-triple, that is, a triple (x, y, z) of positive integers such that y = ax+d and z = bx+2d for some positive integer d. This confirms a conjecture of Landman and Robertson.

Related Organizations

Rutgers, The State University of New Jersey
United States
Massachusetts Institute of Technology
United States

Keywords

Other combinatorial number theory, Ramsey theory, Multiplicative structure; Euclidean algorithm; greatest common divisors

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