*to a man with a hammer, everything looks like a nail*

*Indeed. Since I am teaching a grad course*

*Ramsey theory and its ``Applications''*(I got 24 students, which is more than I thought I would- including around 10 ugrads who are taking it because `all the cool kids are taking it' ) I have been taking the hammer of Ramsey Theory and looking for nails to apply it to. (I'm also using a webiste I found of applications of Ramsey Theory here and an survey article on applications of Ramsey here.)

Thinking about the infinite Ramsey Theorem I came up with this ``application'' :

I

*f p1, p2, p3, ... is an infinite sequence of points in R*

^{n}then there exists a subsequence q1, q2,q3,... such that for each coordinate 1 ≤ i ≤ n the projection onto that coordinate is either (a) strictly incresing, (b) strictly decreasing, or (c) constant.**Proof:**For i< j color (i,j) with one of 3^n colors - indicating for each coordinate i if the ith projection is increaing, decreasing, or constant. The infinite homog set gives you the sequence.

**End of Proof**

**One can also proof this from the Bolzano-Weierstrass theorem (an infinite bounded sequence of points in R**

^{n}has a convergent subsequence). We leave that proof to the reader; however, the proof of BW looks like the proof of the infinite Ramsey Theorem, so I'm not sure if my proof is new or not.

I wanted to look into the BW theorem so I googled "Bolzano-Weierstrass" I think Google knows me better than I know myself since the second hit was https://www.youtube.com/watch?v=dfO18klwKHg which is a Rap Song about the BW theorem (I am a fan of novelty songs, and of math, so does it follow I am a fan of Math Novelty songs. Not sure if its follows, but I AM!)

One of the problems on the HW was

*BW-rap- good, bad, or really bad?*

*Answers were:*

1) Made my ears bleed

2) Lyrics good, singing really bad

3) So bad its good

4) No, just bad.

I'm in the

*Lyrics good/singing is `so bad its good'*camp. The class was okay with the lyrics, but mostly thought it was

*so bad its... bad.*One person thought it was awesome!

I would like to see real rapper perform it on you tube. I doubt I will. Oh well.

On hammers & nails: Probably Abraham Maslow: https://quoteinvestigator.com/2014/05/08/hammer-nail/

ReplyDeleteI double check my answer after seeing this post lol

ReplyDelete