Examples of open problems solved through short proof

Solution 1:

The Stanley-Wilf Conjecture, that the number of permutations of $\{1,2, \ldots, n\}$ avoiding a fixed set of patterns grows at most exponentially with $n$, was open for fifteen or so years until a very short and elegant proof was found by Marcus and Tardos in 2004.

Solution 2:

a semi famous example from number theory, Erdos Proof of Bertrands postulate (paper by Galvin).

In 1845 Bertrand postulated that there is always a prime between n and 2n, and he verified this for $n < 3 \cdot 10^6$ . Tchebychev gave an analytic proof of the postulate in 1850. In 1932, in his first paper, Erdős gave a beautiful elementary proof using nothing more than a few easily verified facts about the middle binomial coefficient. We describe Erdős’s proof and make a few additional comments, including a discussion of how the two main lemmas used in the proof very quickly give an approximate prime number theorem. We also describe a result of Greenfield and Greenfield that links Bertrand’s postulate to the statement that $\{1, . . . , 2n\}$ can always be decomposed into $n$ pairs such that the sum of each pair is a prime.

Solution 3:

I think Hilbert's Basis theorem is a good example. Mathematicians struggled with a more special question than that Hilbert proved. But there was a lot of critic of the proof, that was thought of as religion rather than mathematics.