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A072842 Largest m such that we can partition the set {1,2,...,m} into n subsets with the property that we never have a+b=c for any distinct elements a, b, c in one subset.

%I A072842 #78 Aug 29 2025 09:08:06
%S A072842 2,8,23,66
%N A072842 Largest m such that we can partition the set {1,2,...,m} into n subsets with the property that we never have a+b=c for any distinct elements a, b, c in one subset.
%C A072842 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 }
%C A072842 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]
%C A072842 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
%C A072842 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
%C A072842 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
%C A072842 From _Fred Rowley_, Aug 29 2025: (Start)
%C A072842 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).
%C A072842 The following coloring demonstrates that a(5) >= 207, confirming this number remains an open problem:
%C A072842  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
%C A072842  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
%C A072842  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
%C A072842  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
%C A072842  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
%C A072842  4 4 4 4 3 3 1. (End)
%D A072842 EFNet #math, Jul 23 2002 (can we replace this with a link? - _N. J. A. Sloane_)
%H A072842 R. Ageron, P. Casteras, T. Pellerin, Y. Portella, A. Rimmel and J. Tomasik, <a href="https://arxiv.org/abs/2112.03175">New lower bounds for Schur and weak Schur numbers</a>, arXiv:2112.03175 [math.CO], 2021-2022.
%H A072842 P. Blanchard, F. Harary and R. Reis, <a href="http://www.emis.de/journals/INTEGERS/papers/g7/g7.Abstract.html">Partitions into sum-free sets</a>, Integers: electronic journal of combinatorial number theory, 6. 2006.
%H A072842 P. Bornsztein, <a href="http://dx.doi.org/10.4064/aa101-4-6">An extension of a theorem of Schur</a>, Acta Arithmetica, 101.4 (2001), pp. 395-399.
%H A072842 Dr. Dobb's Journal, <a href="http://www.drdobbs.com/my-enemys-enemy/184411140">Solutions to the "Monopoles" Problem</a>, Dec 01 1999.
%H A072842 S. Eliahou, J. M. Marín, M. P. Revuelta, M. I. Sanz, <a href="http://dx.doi.org/10.1016/j.camwa.2011.11.006">Weak Schur numbers and the search for G.W. Walker’s lost partitions</a>, Computers & Mathematics with Applications, Volume 63, Issue 1, January 2012, Pages 175-182.
%H A072842 Shalom Eliahou, <a href="http://images-archive.math.cnrs.fr/Les-extraordinaires-predictions-du-Reverend-Walker.html">Les extraordinaires prédictions du Révérend Walker</a>, Images des Mathématiques, CNRS, 2017 (in French).
%H A072842 Gordon Hamilton, <a href="http://www.youtube.com/watch?v=nCW4SOJviLo">Grade 2 $1,000,000 Unsolved Problem</a> (2011).
%H A072842 Robert W. Irving, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa25/aa2516.pdf">An extension of Schur's theorem on sum-free partitions</a>, Acta Arithmetica 25 (1973/74), pp. 55-64. (see p. 59ff)
%H A072842 Dmitry Kamenetsky, <a href="https://puzzling.stackexchange.com/questions/90381/paint-numbers-from-1-to-8-with-two-colours">Paint numbers from 1 to 8 with two colours</a>, Puzzling StackExchange, 2019.
%H A072842 Dmitry Kamenetsky, <a href="https://puzzling.stackexchange.com/questions/90388/paint-numbers-from-1-to-23-with-three-colours">Paint numbers from 1 to 23 with three colours</a>, Puzzling StackExchange, 2019.
%H A072842 MathEnJeans, <a href="http://mathenjeans.free.fr/amej/mej_quoi/sujets_10-11/sujets_Calais.pdf">Les tiroirs anti-sommes</a>, 2010-2011 (in French).
%H A072842 D. Robilliard, C. Fonlupt, V. Marion-Poty, Amine Boumaza, <a href="http://www.loria.fr/~boumaza/data/publi/files/fontlupt-2011.pdf">A multilevel Tabu Search with Backtracking for Exploring Weak Schur Numbers</a>, Artificial Evolution, Lecture Notes in Computer Science, Volume 7401, 2012, pp 109-119.
%H A072842 Fred Rowley, <a href="http://math.colgate.edu/∼integers/v59/v59.pdf">New lower bounds for weak Schur partitions</a>, Integers, vol. 21, p. #A59, 2021.
%H A072842 G. W. Walker, <a href="http://www.jstor.org/stable/2306524">Solution to the problem E985</a>, American Mathematical Monthly, Vol. 59 (1952), p. 253.
%F A072842 It is known that 315^((n-1)/5) <= a(n) <= floor(n!*n*e). - Pierre Bornsztein (bornsztein(AT)voila.fr), Sep 02 2003
%F A072842 a(n) < A118771(n), and also a(n) <= A036918(n+1). - _Michel Marcus_, Mar 26 2013
%e A072842 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]
%Y A072842 The requirement that a not equal b is the only difference between these numbers and the Schur numbers A045652.
%Y A072842 Cf. A036918, A118771, A297023.
%K A072842 nonn,more,nice,hard,changed
%O A072842 1,1
%A A072842 Tor G. J. Myklebust (pi(AT)flyingteapot.bnr.usu.edu), Jul 24 2002
%E A072842 Additional comments from _Rob Pratt_ and _Brendan McKay_, Nov 02 2002
%E A072842 More terms from Pierre Bornsztein (bornsztein(AT)voila.fr), Sep 02 2003
%E A072842 Minor additions from Julien de Prabere (jdpbr(AT)aliceadsl.fr), Feb 25 2010
%E A072842 Term a(5) = 196 removed by _Fred Rowley_, Aug 29 2025