Category Archives: Ramsey Theory

RT Part 9; Sparse Ramsey Theory

Posted on 01/29/2012 by Jay Cummings

Well folks, this is it.  This will be the last Ramsey theory-focused post for awhile, I’m guessing.  At least we’re ending on a nice round number! And I hope you agree that we are ending on a nice topic.  It’s … Continue reading →

Posted in Combinatorics, Ramsey Theory | 6 Comments

RT Part 8; Hales-Jewett and how to make Tic-Tac-Toe Interesting

Posted on 01/04/2012 by Jay Cummings

A preliminary note:  I am beginning to lose steam on Ramsey theory.  I want to move on to a few other things and so I will probably gloss over or postpone a few topics that I planned on getting to.  … Continue reading →

Posted in Combinatorics, Ramsey Theory | 1 Comment

RT Part 7; Van Der Waerden’s Theorem

Posted on 11/29/2011 by Jay Cummings

Last time we saw how we can use the technique of focusing to show that however one 2-colors [330], there always exists a 3-term, monochromatic arithmetic progression.  As with last time, let’s abbreviate “-term monochromatic arithmetic progression” as “-TMAP”.  We … Continue reading →

Posted in Combinatorics, Ramsey Theory | 3 Comments

RT Part 6; The Technique of Focusing

Posted on 11/11/2011 by Jay Cummings

Last time I mentioned that we’d now move from Ramsey theory on graphs to Ramsey theory on the naturals.  By popular demand, and to Zach‘s dismay, as decided by a Nebraska Math Department poll last year we will declare that … Continue reading →

Posted in Combinatorics, Ramsey Theory | 7 Comments

RT Part 5; Intro to RT on the Naturals

Posted on 11/04/2011 by Jay Cummings

We have talked about Ramsey theory on graphs.  But what other structures can we do Ramsey theory on?  Well, how about numbers?  Remember that the basic idea behind Ramsey theory is to try to guarantee the existence of some sort … Continue reading →

Posted in Combinatorics, Ramsey Theory, Uncategorized | Leave a comment

RT Part 4; The Happy Ending Theorem

Posted on 10/20/2011 by Jay Cummings

In 1932 a problem was posed to a very talented but not-yet famous group of young mathematicians.  Among a couple others, the group consisted of Eszther Klein, George Szekeres, and Paul Erdős.  Erdős, for instance, was 19 at the time.  … Continue reading →

Posted in Ramsey Theory | 9 Comments

RT Part 3; Hypergraphs and Other Generalizations

Posted on 10/16/2011 by Jay Cummings

Before we move onto colorings of , I thought I’d say one more thing about the generalization of our previous results to hypergraphs and one example of it in action.  This will take two posts.  In this post I wanted … Continue reading →

Posted in Ramsey Theory | 12 Comments

Ramsey Theory Part 2; RT on Infinite, Colorful Graphs

Posted on 10/12/2011 by Jay Cummings

Last time we showed that , the complete graph on vertices has a monochromatic triangle (i.e. a complete graph on vertices) whenever the edges of it are 2-colored.  And we asserted that this can be extended to:  For any natural … Continue reading →

Posted in Ramsey Theory | 9 Comments

Hello! / Intro to Ramsey Theory

Posted on 10/07/2011 by Jay Cummings

If you observe a group of 20 children playing outside, you may notice that you can always find either a group of 4 of them, each of which is friends with the other 3, or you can find a group … Continue reading →

Posted in Ramsey Theory | 8 Comments