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%I #98 Jul 14 2024 08:28:14
%S 1,2,4,9,20,45,112
%N Maximum numbers of cards that would have no SET in an n-attribute version of the SET card game.
%C Or, largest size of an n-dimensional capset (i.e., a subset of (Z/3Z)^n that does not contain any lines {a, a+r, a+2r}). - _Terence Tao_, Feb 20 2009
%C Or, size of maximal cap in the affine geometry AG(n+1,3). - _N. J. A. Sloane_, Oct 25 2014
%H Bela Bajnok, <a href="https://arxiv.org/abs/1705.07444">Additive Combinatorics: A Menu of Research Problems</a>, arXiv:1705.07444 [math.NT], May 2017.
%H Robert A. Bosch, <a href="http://www.mathopt.org/Optima-Issues/optima63.pdf">‘Set’less Collections of SET Cards</a>, 2000.
%H Brink, D. V., 1997, <a href="https://web.archive.org/web/20070519044621/http://omino.com:80/set/">The search for SET</a>. [via WayBackMachine]
%H Benjamin Lent Davis and Diane Maclagan, <a href="http://homepages.warwick.ac.uk/staff/D.Maclagan/papers/set.pdf">The Card Game SET</a>, The Mathematical Intelligencer, Vol. 25:3 (Summer 2003), pp. 33-40.
%H Ernest Davis, <a href="https://cs.nyu.edu/~davise/papers/FunSearch.pdf">Review of Romera-Paredes et al., 14 Dec 2023 Nature paper about FunSearch</a>, January 7, 2024
%H Yves Edel, <a href="http://www.mathi.uni-heidelberg.de/~yves/">Home page</a>.
%H Jordan S. Ellenberg, <a href="https://quomodocumque.wordpress.com/2016/05/13/bounds-for-cap-sets/">Bounds for cap sets</a>, Quomodocumque Blog, May 13 2016.
%H Jordan S. Ellenberg, Dion Gijswijt, <a href="https://arxiv.org/abs/1605.09223">On large subsets of F_q^n with no three-term arithmetic progression</a>, arXiv:1605.09223 [math.CO], 2016.
%H Michael Follett, et al. <a href="http://dx.doi.org/10.1016/j.disc.2014.08.002">Partitions of AG (4, 3) into Maximal Caps</a>, Discrete Math., 337 (2014), 1-8. Preprint: <a href="http://arxiv.org/abs/1302.4703">arXiv:1302.4703</a> [math.CO].
%H Dion Gijswijt, <a href="http://homepage.tudelft.nl/64a8q/papers/set-machazine.pdf">The card game SET: a mathematical challenge</a>, 2016.
%H Dion Gijswijt, <a href="https://www.vvsor.nl/wp-content/uploads/2018/11/STAtOR-2019-2-10-13-Gijswijt.pdf">The Beautiful Mathematics of the Card Game SET</a>, STAtOR, Netherlands Society for Statistics and Operations Research (VVSOR, 2019) Vol. 20, No. 2, 10-13.
%H Guardians of SET, <a href="http://magliery.com/Set/">SET Home Page</a>.
%H Pierre Jalinière, <a href="http://images.math.cnrs.fr/Le-jeu-Set.html">Le jeu Set</a>, Images des Mathématiques, CNRS, 2013.
%H Miriam Melnick, <a href="http://www.setgame.com/sites/default/files/teacherscorner/THE%20JOY%20OF%20SET.pdf">The Joy of SET</a>, May 2011.
%H J. Peebles, <a href="http://www.math.hmc.edu/seniorthesis/archives/2013/jpeebles/jpeebles-2013-thesis-poster.pdf">Cap Set Bounds and Matrix Multiplication</a>, Senior Thesis, Harvey Mudd College, 2013.
%H Ivars Peterson, <a href="http://mathtourist.blogspot.com/2010/07/set-math.html">SET Math</a>.
%H Aaron Potechin, <a href="http://dx.doi.org/10.1007/s10623-007-9132-z">Maximal caps in AG(6, 3)</a>, Designs, Codes and Cryptography, Volume 46, Number 3, March 2008.
%H Bernardino Romera-Paredes, Mohammadamin Barekatain, Alexander Novikov, et al. <a href="https://www.nature.com/articles/s41586-023-06924-6">Mathematical discoveries from program search with large language models</a>, Nature 625, 468-475 (2024).
%H SET card game, <a href="http://www.setgame.com/set">Official web site</a>.
%H Terence Tao, <a href="http://terrytao.wordpress.com/2009/02/05/upper-and-lower-bounds-for-the-density-hales-jewett-problem/">Bounds for the first few density Hales-Jewett numbers, and related quantities</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Set_(game)">Set (game)</a>
%H M. Zabrocki, <a href="http://www.math.yorku.ca/~zabrocki/set/joyofset.pdf">The Joy of SET</a>, 2001.
%F a(n) <= A003142(n).
%F Asymptotically, a(n) = O(3^n/n) and a(n) > (2.21...)^n. - _Terence Tao_, Feb 20 2009
%F Asymptotically, a(n) = o(2.756^n). - _David Radcliffe_, May 30 2016
%Y Cf. A090246, A156989.
%K nonn,hard,more,nice
%O 0,2
%A _Hans Havermann_, Jan 23 2004
%E a(6) from _Terence Tao_, Feb 20 2009
%E Edited by _N. J. A. Sloane_, Feb 21 2009
%E Edited by _Andrey Zabolotskiy_, Mar 01 2024