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The Problem

Let [math]\displaystyle{ [3]^n }[/math] be the set of all length [math]\displaystyle{ n }[/math] strings over the alphabet [math]\displaystyle{ 1, 2, 3 }[/math]. A combinatorial line is a set of three points in [math]\displaystyle{ [3]^n }[/math], formed by taking a string with one or more wildcards [math]\displaystyle{ x }[/math] in it, e.g., [math]\displaystyle{ 112x1xx3\ldots }[/math], and replacing those wildcards by [math]\displaystyle{ 1, 2 }[/math] and [math]\displaystyle{ 3 }[/math], respectively. In the example given, the resulting combinatorial line is: [math]\displaystyle{ \{ 11211113\ldots, 11221223\ldots, 11231333\ldots \} }[/math]. A subset of [math]\displaystyle{ [3]^n }[/math] is said to be line-free if it contains no lines. Let [math]\displaystyle{ c_n }[/math] be the size of the largest line-free subset of [math]\displaystyle{ [3]^n }[/math].

[math]\displaystyle{ k=3 }[/math] Density Hales-Jewett (DHJ(3)) theorem: [math]\displaystyle{ \lim_{n \rightarrow \infty} c_n/3^n = 0 }[/math]

The original proof of DHJ(3) used arguments from ergodic theory. The basic problem to be considered by the Polymath project is to explore a particular combinatorial approach to DHJ, suggested by Tim Gowers.

Useful background materials

Some background to the project can be found here. General discussion on massively collaborative "polymath" projects can be found here. A cheatsheet for editing the wiki may be found here. Finally, here is the general Wiki user's guide

Threads

• (1-199) A combinatorial approach to density Hales-Jewett (inactive)
• (200-299) Upper and lower bounds for the density Hales-Jewett problem (inactive)
• (300-399) The triangle-removal approach (inactive)
• (400-499) Quasirandomness and obstructions to uniformity (inactive)
• (500-599) Possible proof strategies (active)
• (600-699) A reading seminar on density Hales-Jewett (active)
• (700-799) Bounds for the first few density Hales-Jewett numbers, and related quantities (active)

Here is a further list of blog posts related to the Polymath1 project. Here is wordpress's list.

A spreadsheet containing the latest upper and lower bounds for [math]\displaystyle{ c_n }[/math] can be found here. Here are the proofs of our upper and lower bounds for these constants.

We are also collecting bounds for Fujimura's problem, motivated by a hyper-optimistic conjecture.

There is also a chance that we will be able to improve the known bounds on Moser's cube problem.

Here are some unsolved problems arising from the above threads.

Here is a tidy problem page.

Proof strategies

It is natural to look for strategies based on

• Szemerédi's original proof of Szemerédi's theorem.
• Szemerédi's combinatorial proof of Roth's theorem.
• Ajtai-Szemerédi's proof of the corners theorem.
• The density increment method.
• The triangle removal lemma.
• Ergodic-inspired methods.

Bibliography

Density Hales-Jewett

1. H. Furstenberg, Y. Katznelson, “A density version of the Hales-Jewett theorem for k=3“, Graph Theory and Combinatorics (Cambridge, 1988). Discrete Math. 75 (1989), no. 1-3, 227–241.
2. H. Furstenberg, Y. Katznelson, “A density version of the Hales-Jewett theorem“, J. Anal. Math. 57 (1991), 64–119.
3. R. McCutcheon, “The conclusion of the proof of the density Hales-Jewett theorem for k=3“, unpublished.

Behrend-type constructions

1. M. Elkin, "An Improved Construction of Progression-Free Sets ", preprint.
2. B. Green, J. Wolf, "A note on Elkin's improvement of Behrend's construction", preprint.
3. K. O'Bryant, "Sets of integers that do not contain long arithmetic progressions", preprint.

Triangles and corners

1. M. Ajtai, E. Szemerédi, Sets of lattice points that form no squares, Stud. Sci. Math. Hungar. 9 (1974), 9--11 (1975). MR369299
2. I. Ruzsa, E. Szemerédi, Triple systems with no six points carrying three triangles. Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, pp. 939--945, Colloq. Math. Soc. János Bolyai, 18, North-Holland, Amsterdam-New York, 1978. MR519318
3. J. Solymosi, A note on a question of Erdős and Graham, Combin. Probab. Comput. 13 (2004), no. 2, 263--267. MR 2047239