# The polymath project

Tim Gower’s experiment in massively collaborative mathematics is now underway. He’s dubbed it the “polymath project” – if you want to see posts related to the project, I suggest looking here.

The problem to be attacked can be understood (though probably not solved) with only a little undergraduate mathematics. It concerns a result known as the Density Hales-Jewett theorem. This theorem asks us to consider the set $$[ 3 ]^n$$ of all length $$n$$ strings over the alphabet $$1, 2, 3$$. So, for example, $$11321$$ is in $$[3]^5$$. The theorem concerns the existence of combinatorial lines in subsets of $$[3]^n$$. A combinatorial line is a set of three points in $$[3]^n$$, formed by taking a string with one or more wildcards in it, e.g., $$112*1**3\ldots$$, and replacing those wildcards by $$1$$, $$2$$ and $$3$$, respectively. In the example I’ve given, the resulting combinatorial line is:

$$\{ 11211113\ldots, 11221223\ldots, 11231333\ldots \}$$

The Density Hales-Jewett theorem asserts that for any $$\delta > 0$$, for sufficiently large $$n = n(\delta)$$, all subsets of $$[3]^n$$ of size at least $$\delta 3^n$$ contain a combinatorial line,

Apparently, the original proof of the Density Hales-Jewett theorem used ergodic theory; Gowers’ challenge is to find a purely combinatorial proof of the theorem. More background can be found here. Serious discussion of the problem starts here.

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Categorized as polymath1

## 3 comments

1. Azhar Iqbal Kundi says:

LEGENDRE CONJECTURE
This famous conjecture asserts that between “nn” and “(n+1)(n+1)” , n any natural number, there always occurs at least one prime number. Now there are exactely 2n numbers between nn & (n+1)(n+1) ie nn+1, nn+2,….,nn+2n. On the other side Bertrand’s Postulate (which has been proved) gurantee’s that between any m and 2m (m being a natural number) there always exists a prime number. Can these two facts be brought to some common grounds? I think Bertrand Postulate is ultimate key for resolution of Legendre Conecture but how this can be achieved?

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