### 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,

$$GS(n)=\left\{ \begin{array}{ll} 5^k & \mbox{if} \ n=2k \\ 2\cdot 5^k & \mbox{if} \ n=2k+1.\end{array} \right.$$

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MANUSCRIPTISSN: 1855-3974

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