Many people are aware of the Pigeonhole Principle: If we distribute $n+1$ pigeons into $n$ pigeonholes, at least one hole will contain at least two pigeons. However, much fewer are aware of the $\bf{Probabilistic}$ Pigeonhole Principle, which answers the question, 'How many do we usually need?'
Those familiar with the famous birthday problem might have good intuition for this. Let's recall the birthday problem: How many (randomly chosen) people do you need in a room for the probability that some two share a birthday to be more than 1/2? Wikipedia has a nice article on this here. (If this is the first time you've heard the question, it's a great one to think about.) For most people, the answer is quite a surprise, as it is only 23.
Returning to the general problem, if we randomly distribute $n^{1/2}$ pigeons into $n$ pigeonholes, the probability of some overlap approaches 1, as $n \to \infty$.
Let's state that more precisely:
Take any $\epsilon > 0.$ Randomly place $\lfloor n^{1/2 + \epsilon} \rfloor$ pigeons into $n$ holes. (Each is placed independently of the others and with uniform distribution.) Let $\Pr{(n,\epsilon)}$ be the probability that at least two pigeons are placed in the same hole. Then we have, $$\lim_{n \to \infty} \Pr{(n,\epsilon)} = 1.$$ I'm having some trouble proving this, and I also haven't found a proof online. In any case, the proof itself should not be difficult. Here is what I have so far:
Let $m = \lfloor n^{1/2 + \epsilon} \rfloor$. Then we have, $$\Pr{(n,\epsilon)} = 1 - \frac{(n)_m}{n^m},$$ where $(n)_m$ is the falling factorial. i.e. $(n)_m = n(n-1)\times \ldots \times (n-(m-1))$. I tried applying Stirling's formula but after doing some algebraic manipulatoin, I couldn't quite finish the proof. (I also crunched some numbers with Python, providing some numerical evidence.) So, what would be a proof of this?
Here is an extra note on my background to the question. I first became acquainted with it at a math conference when two people, (Piotr Przytycki and another person, whose name I don't recall), gave a short introduction to random (finitely presented) groups. The above principle is used to prove part of a basic result. You can read it in the literature on the bottom of page 31 of the excellent 2005 survey article (on random groups) which Yann Ollivier wrote.
