The Two-Colour Rado Number for the Equation ax + by = (a + b)z
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For relatively prime positive integers a and b, let n = R(a,b) denote the least positive integer such that every 2-colouring of [1, n] admits a monochromatic solution to ax + by = (a + b)z with x, y, z distinct integers. It is known that R(a,b) ≤ 4(a+b) +1. We show that R(a,b) = 4(a+b) +1, except when (a, b) = (3, 4) or (a, b) = (1, 4k) for some k≥1, and R(a,b) = 4(a+b) − 1 in these exceptional cases.
Date issued
2015-04Department
Massachusetts Institute of Technology. Operations Research CenterJournal
Annals of Combinatorics
Publisher
Springer Basel
Citation
Gupta, Swati, J. Thulasi Rangan, and Amitabha Tripathi. “The Two-Colour Rado Number for the Equation Ax + by = (a + B)z.” Annals of Combinatorics 19.2 (2015): 269–291.
Version: Author's final manuscript
ISSN
0218-0006
0219-3094