Geodesic
Schur numbers involving rainbow colorings
Ars Mathematica Contemporanea, Tome 18 (2020) no. 2, pp. 281-288.

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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 thatRS(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) = 5k   if n = 2k;    GS(n) = 2 · 5k   if n = 2k + 1.
DOI : 10.26493/1855-3974.2019.30b
Keywords: Schur numbers, anti-Ramsey numbers, rainbow triangles, Gallai colorings 
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Mark Budden. Schur numbers involving rainbow colorings. Ars Mathematica Contemporanea, Tome 18 (2020) no. 2, pp. 281-288. doi : 10.26493/1855-3974.2019.30b. http://geodesic.mathdoc.fr/articles/10.26493/1855-3974.2019.30b/

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