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A090245 Maximum numbers of cards that would have no SET in an n-attribute version of the SET card game. 4
1, 2, 4, 9, 20, 45, 112 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

It may only be a conjecture that the interpretation in terms of the SET game gives the same sequence for all n as the maximal cap problem. - N. J. A. Sloane, Oct 25 2014

REFERENCES

James Abello (DIMACS Institute, Rutgers University), The majority rule and combinatorial geometry (via the symmetric group), preprint, 2004.

B. Monjardet, Acyclic domains of linear orders: a survey, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 139-160.

LINKS

Table of n, a(n) for n=0..6.

Brink, D. V., 1997, The search for SET

B. L. Davis and D. Maclagan, The Card Game SET

Ben Davis and Diane Maclagan, The Card Game SET, The Mathematical Intelligencer, Vol. 25:3 (Summer 2003), pp. 33-40.

Yves Edel, Home page

Michael Follett, et al. Partitions of AG (4, 3) into Maximal Caps, arXiv preprint arXiv:1302.4703 (2013).

Michael Follett, et al. Partitions of AG (4, 3) into Maximal Caps, Discrete Math., 337 (2014), 1-8.

Guardians of SET, SET Home Page

Pierre Jalinière, Le jeu Set, Images des Mathématiques, CNRS, 2013.

J. Peebles, Cap Set Bounds and Matrix Multiplication, Senior Thesis, Harvey Mudd College, 2013.

Ivars Peterson, SET Math

Ivars Peterson, SET Math

Ivars Peterson, SET Math.

Aaron Potechin, Maximal caps in AG(6, 3), Designs, Codes and Cryptography, Volume 46, Number 3, March 2008.

SET card game, Official web site

Terence Tao, Bounds for the first few density Hales-Jewett numbers, and related quantities

Zabrocki, M., 2001, The Joy of SET

FORMULA

a(n) <= A003142(n).

Asymptotically, a(n) = O(3^n/n) and a(n) > (2.21...)^n. - Terence Tao, Feb 20 2009

CROSSREFS

Cf. A090246, A156989.

Sequence in context: A188460 A111099 A000632 * A006958 A036617 A007902

Adjacent sequences:  A090242 A090243 A090244 * A090246 A090247 A090248

KEYWORD

hard,more,nonn,nice

AUTHOR

Hans Havermann, Jan 23 2004

EXTENSIONS

a(6) sent by Terence Tao, Feb 20 2009

Edited by N. J. A. Sloane, Feb 21 2009

STATUS

approved

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Last modified December 22 12:40 EST 2014. Contains 252356 sequences.