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The Problem

Initially, the basic problem to be considered by the Polymath1 project was to explore a particular combinatorial approach to the density Hales-Jewett theorem for k=3 (DHJ(3)), suggested by Tim Gowers. The original proof of DHJ(3) used arguments from ergodic theory. Fairly soon, the scope of the project expanded and the main aim became that of discovering any combinatorial argument for the theorem. This aim appears to have been achieved but the proof has not yet been fully written up.

Basic definitions

• Algebraic line

• Combinatorial line

• Combinatorial subspace

• Corner

• Density

• Geometric line

• Slice

Useful background materials

Here is some background to the project. There is also a general discussion on massively collaborative "polymath" projects. This is a cheatsheet for editing the wiki. Finally, here is the general Wiki user's guide.

Threads and further problems

• Is massively collaborative mathematics possible? (inactive)
• (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 (inactive)
• (600-699) A reading seminar on density Hales-Jewett (inactive)
• (700-799) Bounds for the first few density Hales-Jewett numbers, and related quantities (inactive)
• (800-849) Brief review of polymath1 (inactive)
• (850-899) DHJ(3): 851-899 (inactive)
• (900-999) DHJ(3): 900-999 (Density Hales-Jewett type numbers) (inactive)
• (1000-1049) Problem solved (probably) (inactive)
• Polymath1 and open collaborative mathematics (active)
• (1050-1099) DHJ(3) and related results: 1050-1099 (active)
• (1100-1199) DHJ(3): 1100-1199 (Density Hales-Jewett type numbers) (active)

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

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 or the Kakeya 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 one of the following:

• 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.
• The Furstenberg-Katznelson argument.
• Use of equal-slices measure.

Related theorems

• Carlson's theorem.
• The Carlson-Simpson theorem.
• Folkman's theorem.
• The Graham-Rothschild theorem.
• The colouring Hales-Jewett theorem.
• The Kruskal-Katona theorem.
• Roth's theorem.
• The IP-Szemerédi theorem.
• Sperner's theorem.
• Szemerédi's regularity lemma.
• Szemerédi's theorem.
• The triangle removal lemma.

All these theorems are worth knowing. The most immediately relevant are Roth's theorem, Sperner's theorem, Szemerédi's regularity lemma and the triangle removal lemma, but some of the others could well come into play as well.

Important concepts related to possible proofs

• Complexity of a set
• Concentration of measure
• Influence of variables
• Obstructions to uniformity
• Quasirandomness

Complete proofs or detailed sketches of potentially useful results

• The multidimensional Sperner theorem
• Line-free sets correlate locally with complexity-1 sets
• Correlation with a 1-set implies correlation with a subspace (Superseded)
• A Fourier-analytic proof of Sperner's theorem
• A second Fourier decomposition related to Sperner's theorem
• A Hilbert space lemma
• A Modification of the Ajtai-Szemerédi argument
• An abstract regularity lemma

Attempts at proofs of DHJ(3)

• An outline of a density-increment argument (ultimately didn't work)
• A second outline of a density-increment argument (seems to be OK but more checking needed)
• Furstenberg-Katznelson argument (very sketchy)
• Austin's proof (very sketchy)
• Austin's proof II (mostly complete, though more explanation and motivation needed)

Generalizing to DHJ(k)

• DHJ(k) implies multidimensional DHJ(k)
• Line free sets correlate locally with dense sets of complexity k-2
• A general partitioning principle

Bibliography

Here is a Bibliography of relevant papers in the field.

How to help out

There are a number of ways that even casual participants can help contribute to the Polymath1 project:

• Expand the bibliography
• Join the metadiscussion thread
• Add some more Ramsey theorems to this wiki; one could hope to flesh out this wiki into a Ramsey theory resource at some point.
• Suggest a logo for this wiki!
• Suggest a way to speed up our genetic algorithm
• Point out places where the exposition could be improved
• Add to this list