Archive for March, 2010

New Papers on Euclidean Ramsey Theory

March 26, 2010

This post is in response to a comment I got asking about new results in Ramsey theory especially the newer ones.

One thing I would recommend is reading the papers of Ronald Graham on Euclidean Ramsey theory and looking at the bibliographies of these papers and looking at the papers there. His papers are here.
I think there is a preprint on Euclidean Ramsey theory that is submitted. It is a survey on open problems in the field.

Some other recent papers:

“A Note on Euclidean Ramsey Theory”
by James H. Schmerl
Discrete and Computational Geometry
Volume 38, Number 1 / July, 2007
Springer New York

“An Improvement to “A Note on Euclidean Ramsey Theory”
by James H. Schmerl
Discrete and Computational Geometry
Volume 43, Number 2 / March, 2010

“On the chromatic numbers of small-dimensional Euclidean spaces”
by A.B. Kupavskii1, and A.M. Raigorodskii
Electronic Notes
in Discrete Mathematics
Volume 34, 1 August 2009, Pages 435-439

“The chromatic number of the plane: The bounded case”
by Clyde P. Kruskala,
Journal of Computer and System Sciences
Volume 74, Issue 4, June 2008, Pages 598-627

and also one by me:

“All regular polytopes are Ramsey”
by Kristal Cantwell
Journal of Combinatorial Theory, Series A
Volume 114, Issue 3, April 2007, Pages 555-562

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A new Thread

March 24, 2010

There is a new thread for Polymath5. It looks like Klas Markstrom is getting close to a proof that for a large enough N any sequence of length greater than one will have discrepancy at least 3. I think there is one last case that is being analyzed. If everything works out I think N will be around 2000.

Polymath5

March 14, 2010

There is a new thread for polymath5. The new thread is more of a place holder. It has material but that material is not yet applied to the problem. It should be completed soon. Let me update this: The post is now complete and it now outlines a new approach to the problem.

New ideas in Polymath5

March 2, 2010

There is a new thread for Polymath5. There some new ideas involving among other things quadratic forms which seem to make the problem more tractable.

Let me update this there is another thread. There are two methods used one involves semidefinite programming. The other method which is completely different involves constructing a polynomial over finite field.