# 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 of all length strings over the alphabet . So, for example, is in . The theorem concerns the existence of *combinatorial lines* in subsets of . A combinatorial line is a set of three points in , formed by taking a string with one or more wildcards in it, e.g., , and replacing those wildcards by , and , respectively. In the example I’ve given, the resulting combinatorial line is:

The Density Hales-Jewett theorem asserts that for any , for sufficiently large , all subsets of of size at least 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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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?