A003142 -id:A003142 - cp's OEIS frontend

A090245 Maximum numbers of cards that would have no SET in an n-attribute version of the SET card game.

1, 2, 4, 9, 20, 45, 112

Offset: 0

Hans Havermann, Jan 23 2004

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

Or, size of maximal cap in the affine geometry AG(n+1,3). - N. J. A. Sloane, Oct 25 2014

Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017.
Robert A. Bosch, ‘Set’less Collections of SET Cards, 2000.
D. V. Brink, The search for SET, 1997. [via WayBackMachine]
Benjamin Lent Davis and Diane Maclagan, The Card Game SET, The Mathematical Intelligencer, Vol. 25:3 (Summer 2003), pp. 33-40.
Ernest Davis, Review of Romera-Paredes et al., 14 Dec 2023 Nature paper about FunSearch, January 7, 2024
Yves Edel, Caps, 2010.
Yves Edel, Sandy Ferret, Ivan Landjev, Leo Storme, The classification of the largest caps in AG(5,3), Journal of Combinatorial Theory A, 99 (2002), 95-110.
Jordan S. Ellenberg, Bounds for cap sets, Quomodocumque Blog, May 13 2016.
Jordan S. Ellenberg and Dion Gijswijt, On large subsets of F_q^n with no three-term arithmetic progression, arXiv:1605.09223 [math.CO], 2016.
Michael Follett, et al. Partitions of AG (4, 3) into Maximal Caps, Discrete Math., 337 (2014), 1-8. Preprint: arXiv:1302.4703 [math.CO].
Dion Gijswijt, The card game SET: a mathematical challenge, 2016.
Dion Gijswijt, The Beautiful Mathematics of the Card Game SET, STAtOR, Netherlands Society for Statistics and Operations Research (VVSOR, 2019) Vol. 20, No. 2, 10-13.
Guardians of SET, SET Home Page.
Pierre Jalinière, Le jeu Set, Images des Mathématiques, CNRS, 2013.
Miriam Melnick, The Joy of SET, May 2011.
J. Peebles, Cap Set Bounds and Matrix Multiplication, Senior Thesis poster (see also the thesis itself), Harvey Mudd College, 2013.
Ivars Peterson, SET Math.
Aaron Potechin, Maximal caps in AG(6, 3), Designs, Codes and Cryptography, Volume 46, Number 3, March 2008.
Bernardino Romera-Paredes, Mohammadamin Barekatain, Alexander Novikov, et al. Mathematical discoveries from program search with large language models, Nature 625, 468-475 (2024).
SET card game, Official web site.
Terence Tao, Bounds for the first few density Hales-Jewett numbers, and related quantities.
Wikipedia, Set (card game)
M. Zabrocki, The Joy of SET, 2001.

Cf. A090246, A156989.

a(n) <= A003142(n).
Asymptotically, a(n) = O(3^n/n) and a(n) > (2.21...)^n. - Terence Tao, Feb 20 2009
Asymptotically, a(n) = o(2.756^n). - David Radcliffe, May 30 2016

a(6) from Terence Tao, Feb 20 2009
Edited by N. J. A. Sloane, Feb 21 2009
Edited by Andrey Zabolotskiy, Mar 01 2024

A156989 Largest size of a subset of {1,2,3}^n that does not contain any combinatorial lines (i.e., strings formed by 1, 2, 3, and at least one instance of a wildcard x, with x then substituted for 1, 2, or 3, e.g. 12x3x gives the combinatorial line 12131, 12232, 12333.)

Original entry on oeis.org

1, 2, 6, 18, 52, 150, 450

Offset: 0

Terence Tao, Feb 20 2009

The density Hales-Jewett theorem implies that a(n) = o(3^n). a(n) is studied further in the polymath1 project, see link below.

			For n=2, one example that shows a(2) is at least 6 is { 11, 13, 22, 23, 31, 32 }.

Thomas Bloom, Problem 171, Erdős Problems.
Hillel H. Furstenberg and Yitzhak Katznelson, A density version of the Hales-Jewett theorem for k=3, Graph Theory and Combinatorics (Cambridge, 1988). Discr. Math. 75 (1989) no. 1-3, 227-241.
Hillel H. Furstenberg and Yitzhak Katznelson, A density version of the Hales-Jewett theorem, J. Anal. Math. 57 (1991), 64-119.
Kevin O'Bryant, Sets of natural numbers with proscribed subsets, arXiv:1410.4900 [math.NT], 2014-2015.
Kevin O'Bryant, Sets of Natural Numbers with Proscribed Subsets, J. Int. Seq. 18 (2015) # 15.7.7.
D. H. J. Polymath, Density Hales-Jewett and Moser numbers, arXiv:1002.0374 [math.CO], 2010.
Polymath1 Project, Wiki Main Page
Terence Tao, Bounds for the first few density Hales-Jewett numbers, and related quantities
Terence Tao, Erdős problem database, see no. 171.

Bounded below by A003142. Cf. A000244, A090245.

cp's OEIS Frontend

A090245 Maximum numbers of cards that would have no SET in an n-attribute version of the SET card game.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions