- I didn't know the classic upper bounds on the higher Ramsey Numbers until 2 months ago (note that I have been studying Ramsey Theory with a passionfor about 8 years). Upon hearing the theorem 8 years ago I came up with a proof of my own (which had TOWER bounds). I later forgot that I came up with it myself and thought it was the classic proof. I recently saw a reference that said it had double-exp bounds. This confused me so I looked it up and then realized that I never knew the classic proof! Now I do. And I have written up both my proof and the classic proof here. This is odd--- it is common to hear a proof and then later think you came up with it on your own. This was the opposite- I came up with a proof on my own and assumed I had heard it somewhere. I don't quite know what to make of this- I am happy that I came up with a new proof, but it gives worse bounds and is has no advantage whatsoever. (Both proofs are of about the same difficulty, though you can read them and see what you think.)
- I know of two professors, one in Cryptography, and one in Parallelism, who did not know that every planar graph has a vertex of degree at most 5 until I mentioned it casually in a lecture. One said I missed the lecture on planarity in my graph theory class. While I believe this to be true I was surprised that he hadn't heard it somewhere else. Also, there is a point where you should stop thinking of knowledge in terms of courses you took. The other said Why should I care? My answer: it gives an easy proof that all planar graphs are 6-colorable, and it is the starting point for a fairly easy proof that all planar graphs are 5-colorable. He was not impressed.
- A Complexity theorist I knew did not know that if you are operating mod a product of L primes, a number can have 2L square roots.
- An Applied Math Professor I knew did not know that the reals were uncountable until I told him. He was a very practical person and didn't seem to care, calling it a mathematical trick. Oddly enough he used the term measure 0 sometimes.
- Someone I knew who worked in Probabilistic Learning Theory did not know what variance is.
- I knew a grad student in applied math who did not know that eπ i=-1.
- There are several complexity theorists who do not know there is a c.e. set that is not Turing Equivalent to HALT and not decidable. When I tell them there is such they say Oh, just like Ladner's Theorem. This is true but sort-of backwards.
I would have thought that a planar graph having a vertex of degree at most 5 is in both categories, but perhaps I am wrong. I would have thought that knowing the reals are uncountable is in both categories, but here I am RIGHT.
So I ask you readers: name something you or someone you know found out far later in their academic life then is expected.