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.)

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

Offset: 0

Author

Terence Tao, Feb 20 2009

Keywords

• hard
• more
• nonn

Comments

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

Examples

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

Links

• H. Furstenberg, Y. 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.
• H. Furstenberg and Y. Katznelson, A density version of the Hales-Jewett theorem, J. Anal. Math. 57 (1991), 64-119.
• K. O'Bryant, Sets of natural numbers with proscribed subsets, arXiv:1410.4900 [math.NT], 2014-2015.
• K. 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]
• Polymath1 Project, Wiki Main Page
• Terence Tao, Bounds for the first few density Hales-Jewett numbers, and related quantities

Crossrefs

Bounded below by A003142. Cf. A000244, A090245.