In a blog post, rjlipton proves that when . I posted a comment trying to strengthen the first inequality to That comment is almost unreadable because (1) I don't know how to use LaTeX on wordpress, (2) especially if I get only one try (can't edit). Here, I don't have such problems.
Proof. Consider a hashtable with buckets and items. There are ways to distribute the items, out of which have no collision. The probability of having some collision is . Now suppose is a random variable whose value says how many collisions there are. So, The inequality comes from the first moment principle. The expected number of collisions increases with , so we just look at the case . There are possible collisions, each happening with probability . Thus, Putting the previous two displayed inequalities together, we get and finally
Update: Stefan Kiefer pointed out that the tight bounds are
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