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