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 R S ( 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 $$RS(n)=\floor{\log _2(n)}+2, \quad \mbox{for all } n\ge 3.$$ Second, we consider the Gallai-Schur number G S ( n ) , defined to be the least natural number such that every n -coloring of {1, 2, …,  G S ( 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 , $$GS(n)=\left\{ \begin{array}{ll} 5^k & \mbox{if} \ n=2k \\ 2\cdot 5^k & \mbox{if} \ n=2k+1.\end{array} \right.$$