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equation x + y = z has a monochrome solution with x \ne y

Source: 2017 Romania JBMO TST 2.2

June 27, 2020
combinatoricsColoring

Problem Statement

Determine the smallest positive integer nn such that, for any coloring of the elements of the set {2,3,...,n}\{2,3,...,n\} with two colors, the equation x+y=zx + y = z has a monochrome solution with x≠yx \ne y.
(We say that the equation x+y=zx + y = z has a monochrome solution if there exist a,b,ca, b, c distinct, having the same color, such that a+b=ca + b = c.)
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