A possible Polymath problem

There is a post about possible Polymath projects related to Littlewoods conjecture here:

http://gowers.wordpress.com/2009/11/17/problems-related-to-littlewoods-conjecture-2/

Polynomial DHJ

There is a new post about Polynomial DHJ. It is about a possible future polymath project. It is problem number two mentioned in the earlier list of possible Polymath projects.
The question is there a generalization of DHJ that implies both DHJ and the polynomial Szemerédi’s theorem. The coloring result of Bergelson and McCutcheon that simultaneously generalizes the Hales-Jewett theorem and van der Waerden’s theorem is used to form a similar density conjecture and then a variant of that is proposed as a Polymath project. the post is here:

http://gowers.wordpress.com/2009/11/14/the-first-unknown-case-of-polynomial-dhj/

The origin of life

There is a post here:

http://gowers.wordpress.com/2009/11/07/polymath-and-the-origin-of-life/

on the origin of life as a possible polymath project.

LaTeX support for Mathoverflow

Math over flow now has LaTex support! See this thread for more information:

http://meta.mathoverflow.net/discussion/27

New thread for deterministic way to find primes

There is a new thread for polymath4, the url is:

http://polymathprojects.org/2009/10/27/research-thread-v-determinstic-way-to-find-primes/

New version of Polymath1 paper

There is a new version of the Polymath1 paper “A new proof of the density Hales-Jewett theorem.”

Here is the url:

http://www.cs.cmu.edu/~odonnell/papers/dhj.pdf

Let me update this. The paper is now on arXiv at

http://arxiv.org/abs/0910.3926

Site for answering math questions

There is a site for answering questions related to mathematics. The url is here:

http://mathoverflow.net/

As I understand it the question should be of interest to at least one mathematician. People are rated in regards to their responses and with higher ratings come higher privileges. There is more information at the site.

Nature article on polymath

There is a article on polymath here:

http://www.nature.com/nature/journal/v461/n7266/full/461879a.html

Radon related problem 8

Let us generalize this problem to r sets and dimension d:

We have r(d+1) -1 points on the plane. We want to show that we can divide them into r-1 sets of d+1 points and one set of d-2 such that the intersection of the convex hulls of all r sets is not empty.

We take (r-1)*(d+1) +1 of the points and apply Tverberg’s theorem.
It gives us r disjoint nonempty sets which together contain all (r-1)*(d+1) +1 points. The intersection of the convex hulls of these sets is nonempty. Let one of the points in the intersection of the convex hulls be x.

If any of the r sets are bigger than d+1 then by Caratheodory’s theorem there is a d+1 subset that contains x. We take that as the new set in place of the old set and add the additional points to sets less than d+1 making sure none of the additions causes a set to be greater than d+1. We repeat this process untill all of the sets are three or less. Then we add the remaining d-1 points to a sets with d points or less again making sure no set goes over d+1. The result is (r)*(d+1) +1 points divided into r sets of d+1 or less. The only way to do this is r-1 sets of d+1 and one set of d points which gives the desired theorem.

We can add more points and keep the distribution even since adding a point to a set of points increases the convex hull so the intersection of convex hulls still contains x. So we have the following if we have r(d+1) -1 points or more we can divide them into r nonempty sets, the size of each set differing by at most one element from each other set such that the intersection of the convex hulls of the sets is nonempty.

Polymath3 update

There is a new thread for Polymath3:

http://gilkalai.wordpress.com/2009/10/06/the-polynomial-hirsch-conjecture-discussion-thread-continued/