In “How To Solve It”, George Pólya writes:
“There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain
theorem, and I said it is not proved and it may be difficult. Von
Neumann didn’t say anything but after five minutes he raised his hand.
When I called on him he went to the blackboard and proceeded to write
down the proof. After that I was afraid of von Neumann.” [2nd ed.
(1957), p. xv]
Could someone tell me please what was that theorem which von Neumann proved?
John von Neumann studied at Zürich during in Ph.D. and got Ph.D at the age of 22, in 1925. Before that year, Neumann published four papers (see here).
The first one, published in 1922, is about roots of polynomials. The paper does not mention Pólya, but Pólya is known to have worked on ths subject.
The second paper is about the ordinals, the third about set theory which seems less likely. The fourth (in Hungarian) is about sequences and could contain the theorem, but I have not been able to access this article.
Of course, there is the possibility that von Neumann never wrote a paper about this theorem.