As shown by Schur (1916), the Schur number 
satisfies
 |
for 
,
2, ..., where 
is a Ramsey number.
Schur's Ramsey Theorem
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References
Fredricksen, H. and Sweet, M. M. "Symmetric Sum-Free Partitions and Lower Bounds for Schur Numbers." Elec. J. Combin. 7, No. 1, R32, 1-9, 2000. https://doi.org/10.37236/1510.Schur, I. "Über die Kongruenz
mod 
." Jahresber. Deutsche Math.-Verein. 25,
114-116, 1916.Referenced on Wolfram|Alpha
Schur's Ramsey TheoremCite this as:
Weisstein, Eric W. "Schur's Ramsey Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SchursRamseyTheorem.html