Schur numbers involving rainbow colorings

Mark Budden


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
$$RS(n)=\floor{\log _2(n)}+2, \quad \mbox{for all } n\ge 3.$$
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,
$$GS(n)=\left\{ \begin{array}{ll} 5^k & \mbox{if} \ n=2k \\ 2\cdot 5^k & \mbox{if} \ n=2k+1.\end{array} \right.$$


Anti-Ramsey numbers, rainbow triangles, Gallai colorings

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ISSN: 1855-3974

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