Ian Parberry once told me (though I doubt he originated it- The first link I found says it was Mark Twain)
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'' :
If p1, p2, p3, ... is an infinite sequence of points in Rn 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 Rn 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?
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.