

A090245


Maximum numbers of cards that would have no SET in an nattribute version of the SET card game.


6




OFFSET

0,2


COMMENTS

Or, largest size of an ndimensional 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, following a conversation with James Abello.
Or, maximum number of tokens that can placed on an ndimensional 3x3x...x3 bingo card, without creating a bingo (disregarding diagonals).  Philippe Beaudoin, Jun 01 2016


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. 139160.


LINKS

Table of n, a(n) for n=0..6.
Brink, D. V., 1997, The search for SET. [Dead link]
Benjamin Lent Davis and Diane Maclagan, The Card Game SET, The Mathematical Intelligencer, Vol. 25:3 (Summer 2003), pp. 3340.
Yves Edel, Home page.
Jordan S. Ellenberg, Bounds for cap sets, Quomodocumque Blog, May 13 2016.
Jordan S. Ellenberg, Dion Gijswijt, On large subsets of F_q^n with no threeterm arithmetic progression, arXiv:1605.09223 [math.CO], 2016.
Michael Follett, et al. Partitions of AG (4, 3) into Maximal Caps, Discrete Math., 337 (2014), 18. Preprint: arXiv:1302.4703 [math.CO].
Guardians of SET, SET Home Page.
Pierre JaliniÃ¨re, Le jeu Set, Images des MathÃ©matiques, CNRS, 2013.
Miriam Melnick, The Joy of SET, May 2011.
J. Peebles, Cap Set Bounds and Matrix Multiplication, Senior Thesis, 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.
SET card game, Official web site.
Terence Tao, Bounds for the first few density HalesJewett 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
Asymptotically, a(n) = o(2.756^n).  David Radcliffe, May 30 2016


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



