The Schur number
is the largest integer
for which the interval
can be partitioned into sum-free sets (Fredricksen and Sweet 2000). is guaranteed to exist for each by Schur's problem. Note
the definition of the Schur number as the smallest number for which such a partition does not exist
is also prevalent in the literature (OEIS A030126;
Fredricksen and Sweet 2000).
Schur (1916) gave the lower bound
(1)
which is sharp for ,
2, and 3 (Guy 1994). The Schur numbers also satisfy the inequality
(2)
for and some constant (Abbott and Moser 1966, Abbott and Hanson 1972, Exoo 1994).
Schur's Ramsey theorem also shows that
(3)
where
is a Ramsey number. The first few Schur numbers
are 1, 4, 13, 44,
(Fredricksen 1979), ,
, ... (OEIS A045652;
Fredricksen and Sweet 2000). is due to Baumert (Baumert 1965, Abbott and Hanson 1972),
the lower bound on
is due to Exoo (1994), and the lower limits on and are due to Fredricksen and Sweet (2000).
Abbott, H. L. and Hanson, D. "A Problem of Schur ad its Generalizations." Acta Arith.20, 175-187, 1972.Abbott,
H. L. and Moser, L. "Sum-Free Sets of Integers." Acta Arith.11,
392-396, 1966.Baumert, L. D. and Golomb, S. W. "Backtrack
Programming." J. Ass. Comp. Machinery12, 516-524, 1965.Beutelspacher,
A. and Brestovansky, W. "Generalized Schur Numbers." In Combinatorial
Theory. Proceedings of a Conference Held at Schloss Rauischholzhausen, May 6-9, 1982.
Berlin: Springer-Verlag, pp. 30-38, 1982.Exoo, G. "A Lower
Bound for Schur Numbers and Multicolor Ramsey Numbers of ." Electronic J. Combinatorics1, No. 1,
R8, 1-3, 1994. http://www.combinatorics.org/Volume_1/Abstracts/v1i1r8.html.Fredricksen,
H. "Schur Numbers and the Ramsey Numbers ." J. Combin. Theory Ser. A27,
376-377, 1979.Fredricksen, H. and Sweet, M. M. "Symmetric
Sum-Free Partitions and Lower Bounds for Schur Numbers." Electronic J. Combinatorics7,
No. 1, R32, 1-9, 2000. http://www.combinatorics.org/Volume_7/Abstracts/v7i1r32.html.Guy,
R. K. "Schur's Problem. Partitioning Integers into Sum-Free Classes"
and "The Modular Version of Schur's Problem." §E11 and E12 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 209-212,
1994.Radziszowski, S. P. "Small Ramsey Numbers." Electronic
J. Combin.1, DS1 1-29, Rev. Jul. 5, 1999. http://www.combinatorics.org/Surveys/.Schur,
I. "Über die Kongruenz (mod )." Jahresber. Deutsche Math.-Verein.25,
114-116, 1916.Sloane, N. J. A. Sequences A030126
and A045652 in "The On-Line Encyclopedia
of Integer Sequences."Whitehead, E. G. "The Ramsey Number
." Disc. Math.4,
389-396, 1973.
is the largest integer 
for which the interval ![[1,n]](/images/equations/SchurNumber/Inline3.svg)
can be partitioned into 
sum-free sets (Fredricksen and Sweet 2000). 
is guaranteed to exist for each 
by Schur's problem. Note
the definition of the Schur number as the smallest number 
for which such a partition does not exist
is also prevalent in the literature (OEIS A030126;
Fredricksen and Sweet 2000).
,
2, and 3 (Guy 1994). The Schur numbers also satisfy the inequality
and some constant 
(Abbott and Moser 1966, Abbott and Hanson 1972, Exoo 1994).
Schur's Ramsey theorem also shows that
is a Ramsey number. The first few Schur numbers
are 1, 4, 13, 44, 
(Fredricksen 1979), 
,
, ... (OEIS A045652;
Fredricksen and Sweet 2000). 
is due to Baumert (Baumert 1965, Abbott and Hanson 1972),
the lower bound on 
is due to Exoo (1994), and the lower limits on 
and 
are due to Fredricksen and Sweet (2000).
." Electronic J. Combinatorics 1, No. 1,
R8, 1-3, 1994. 
." J. Combin. Theory Ser. A 27,
376-377, 1979.Fredricksen, H. and Sweet, M. M. "Symmetric
Sum-Free Partitions and Lower Bounds for Schur Numbers." Electronic J. Combinatorics 7,
No. 1, R32, 1-9, 2000. 
(mod 
)." Jahresber. Deutsche Math.-Verein. 25,
114-116, 1916.Sloane, N. J. A. Sequences 
." Disc. Math. 4,
389-396, 1973.