OFFSET
1,1
COMMENTS
The fourth term is at least 66 (Ernst Munter), from { 24 26 27 28 29 30 31 32 33 36 37 38 39 41 42 44 45 46 47 48 49 } { 9 10 12 13 14 15 17 18 20 54 55 56 57 58 59 60 61 62 } { 1 2 4 8 11 16 22 25 40 43 53 66 } { 3 5 6 7 19 21 23 34 35 50 51 52 63 64 65 }
Another set of subsets can be described with this sequence of digits (among 8238): 112122213313333333232124144444144422244144441444412223333333331222 (where each digit represents a subset) The fifth term is at least 195 and can be built with the previous sequence, 515, then 66 digits 5 and finally the sequence 122133333333312224144441444222441444444441422213333133331222. I'd like to see a 196-digit sequence. [Julien de Prabere]
Actually a(5)=196 was given by Walker without proof. But Eliahou et al. give an example of such a partition, so a(5) >= 196. And Robilliard et al. give an example for n=6 with [1..574], so a(6) >= 574. - Michel Marcus, Mar 26 2013
To clarify: a(1)-a(4) are known. a(5) = 196 was claimed by Walker but no proof is known, though the value seems likely to be correct. - Charles R Greathouse IV, Jun 13 2013
The best known lower bounds for the next terms: a(6) >= 582, a(7) >= 1740, a(8) >= 5201, a(9) >= 15596. See link to Eliahou's 2017 article. - Dmitry Kamenetsky, Oct 20 2019
From Fred Rowley, Aug 29 2025: (Start)
New lower bounds a(6) >= 642, a(7) >= 2146, a(8) >= 6976 and beyond were established by Rowley in 2021 (see link). Improved lower bounds a(6) >= 646, a(9) >= 22536 and a(10) >= 71256 and beyond were established in 2022 by Ageron et al (see link).
The following coloring demonstrates that a(5) >= 207, confirming this number remains an open problem:
1 2 1 3 1 2 1 3 1 2 1 4 1 2 1 3 3 2 1 4 1 3 3 4 4 4 4 4 4 4 4 3 4 4 4 2 3 1 5 2
2 3 3 2 2 1 5 2 2 1 5 2 2 1 5 2 2 1 5 2 2 1 5 2 2 1 5 2 2 1 5 2 2 1 5 3 3 4 4 4
4 4 4 4 4 4 4 4 4 1 5 5 5 1 5 3 3 1 5 5 5 1 5 5 5 1 5 5 5 1 5 5 5 1 5 5 5 1 5 5
5 1 5 5 5 1 5 5 5 1 3 3 5 1 5 5 5 1 4 4 4 4 4 4 4 4 4 4 4 4 3 3 5 1 2 2 5 1 2 2
5 1 2 2 5 1 2 2 5 1 2 2 5 1 2 2 5 1 2 2 5 1 2 2 3 3 2 2 5 1 3 3 4 4 4 4 4 4 4 4
4 4 4 4 3 3 1. (End)
a(6) >= 650 (see Rodrigo (2026)). - Sebastián E. Rodrigo, Jun 10 2026
REFERENCES
EFNet #math, Jul 23 2002 (can we replace this with a link? - N. J. A. Sloane)
LINKS
Romain Ageron, Paul Casteras, Thibaut Pellerin, Yann Portella, Arpad Rimmel, and Joanna Tomasik, New lower bounds for Schur and weak Schur numbers, arXiv:2112.03175 [math.CO], 2021-2022.
Peter F. Blanchard, Frank Harary, and Rogério Reis, Partitions into sum-free sets, Integers 6 (2006), Article A7.
P. Bornsztein, An extension of a theorem of Schur, Acta Arithmetica, 101.4 (2001), 395-399.
Dr. Dobb's Journal, Solutions to the "Monopoles" Problem, Dec 01 1999.
Shalom Eliahou, J. M. Marín, M. P. Revuelta, and M. I. Sanz, Weak Schur numbers and the search for G. W. Walker’s lost partitions, Computers & Math. with Appl. 63(1) (January 2012), 175-182.
Shalom Eliahou, Les extraordinaires prédictions du Révérend Walker, Images des Mathématiques, CNRS, 2017 (in French).
Gordon Hamilton, Grade 2 $1,000,000 Unsolved Problem (2011).
Robert W. Irving, An extension of Schur's theorem on sum-free partitions, Acta Arithmetica 25 (1973/74), 55-64. See p. 59ff.
Dmitry Kamenetsky, Paint numbers from 1 to 8 with two colours, Puzzling StackExchange, 2019.
Dmitry Kamenetsky, Paint numbers from 1 to 23 with three colours, Puzzling StackExchange, 2019.
MathEnJeans, Les tiroirs anti-sommes, 2010-2011 (in French).
Denis Robilliard, Cyril Fonlupt, Virginie Marion-Poty, and Amine Boumaza, A multilevel Tabu Search with Backtracking for Exploring Weak Schur Numbers, Artificial Evolution, Lect. Notes Comp. Sci. 7401, (2012), 109-119.
Sebastián Rodrigo, A new lower bound for the sixth weak Schur number: WS(6) >= 650, Zenodo, 2026.
Fred Rowley, New lower bounds for weak Schur partitions, Integers 21 (2021), Article A59.
Alexander Towell, Rado Numbers for x + y = k*z: Computation, Closed-Form Formula, and Complete Proof, 2026. See pp. 2, 13.
G. W. Walker, Solution to the problem E985, Amer. Math. Monthly 59 (1952), 253.
FORMULA
It is known that 315^((n-1)/5) <= a(n) <= floor(n!*n*e). - Pierre Bornsztein (bornsztein(AT)voila.fr), Sep 02 2003
EXAMPLE
a(2) = 8 because we may partition the set {1, 2, ..., 8} into {1, 2, 4, 8} and {3, 5, 6, 7} with the desired property, and this is the unique solution; attempting to add 9 to either will produce a set with the property that a+b=c for some a,b,c (1+8=9 or 3+6=9). [Corrected by Julien de Prabere, Dec 17 2009]
CROSSREFS
KEYWORD
nonn,more,nice,hard
AUTHOR
Tor G. J. Myklebust (pi(AT)flyingteapot.bnr.usu.edu), Jul 24 2002
EXTENSIONS
Additional comments from Rob Pratt and Brendan McKay, Nov 02 2002
More terms from Pierre Bornsztein (bornsztein(AT)voila.fr), Sep 02 2003
Minor additions from Julien de Prabere (jdpbr(AT)aliceadsl.fr), Feb 25 2010
Term a(5) = 196 removed by Fred Rowley, Aug 29 2025
STATUS
approved