Schur numbers involving rainbow colorings

Abstract

In this paper, we introduce two different generalizations of Schur numbers that involve rainbow colorings. Motivated by well-known generalizations of Ramsey numbers, we first define the rainbow Schur number RS(n) to be the minimum number of colors needed such that every coloring of {1, 2, …, n}, in which all available colors are used, contains a rainbow solution to a + b = c. It is shown that

RS(n) = ⌊log₂(n)⌋ + 2,   for all n ≥ 3.

Second, we consider the Gallai-Schur number GS(n), defined to be the least natural number such that every n-coloring of {1, 2, …, GS(n)} that lacks rainbow solutions to the equation a + b = c necessarily contains a monochromatic solution to this equation. By connecting this number with the n-color Gallai-Ramsey number for triangles, it is shown that for all n ≥ 3,

GS(n) = 5^k   if n = 2k;    GS(n) = 2 · 5^k   if n = 2k + 1.