Short Egyptian fractions

We prove a conjecture of Erdős: for every sufficiently large integer b, every rational number $a/b$ with $1\le a<b$ is a sum of $O(\log \log b)$ distinct positive unit fractions, with an absolute implied constant. This order is best possible when the numerator varies. We also show that both the number of expansions of 1 with exactly k distinct terms and the least integer at least 2 that never occurs as a denominator in such an expansion grow doubly exponentially in k: their double logarithms have order k.