# Polymath1

## 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 forked. Two projects emerged. The first project, "New Proof", had as its aim that of discovering any combinatorial argument for the theorem. The second project, "Low Dimensions", aimed to calculate precise bounds on density Hales-Jewett numbers and Moser numbers for low dimensions n = 3, 4, 5, 6, 7, etc. Both projects appear to have been successful, and are in the writing-up stage.

## Write-up repositories

Write-up page for the "New Proof" project. The most recent draft is here, tentatively titled "A new proof of the density Hales-Jewett theorem".

Write-up page for the "Low Dimensions" project. The most recent draft is here, tentatively titled "Density Hales-Jewett and Moser numbers in low dimensions".

## 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.

A spreadsheet containing the latest upper and lower bounds for $c_n$ 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.

## 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.

## 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