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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

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. Here is a python script which can help convert sizeable chunks of LaTeX into wiki-tex. Finally, here is the general Wiki user's guide.

Threads and further problems

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]c_n[/math] can be found here. Here are the proofs of our upper and lower bounds for these constants, as well as the counterparts for higher k.

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

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.

Writeup

Here is a wiki-style outline of first paper. Here are (the beginnings of) TeX files for first paper.

Here is a wiki-style outline of second paper.

Proof strategies

It is natural to look for strategies based on one of the following:

Related theorems

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

Complete proofs or detailed sketches of potentially useful results

Attempts at proofs of DHJ(3)

Generalizing to DHJ(k)

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
  • Join the open discussion thread (about these mathematical problems; not about the open collaboration), and participate in the polls.
  • 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
  • Jump in to the technical discussion; find some nice new angles to the discussed problems.
  • Add to this list