  An Egyptian fraction is a sum of distinct unit fractions (reciprocals of
positive integers). We show that every rational number has Egyptian fraction
representations where the number of terms is of the same order of magnitude as
the largest denominator, improving a result from an earlier paper to
best-possible form. We also settle, in best-possible form, the related problem
of how small M_t(r) can be such that there is an Egyptian fraction
representation of r with exactly t terms, the denominators of which are all at
most M_t(r). We also consider the following problems posed by Erdos and Graham:
the set of integers that cannot be the largest denominator of an Egyptian
fraction representation of 1 is infinite - what is its order of growth? How
about those integers that cannot be the second-largest (third-largest, etc.)
denominator of such a representation? In the latter case, we show that only
finitely many integers cannot be the second-largest (third-largest, etc.)
denominator of such a representation; while in the former case, we show that
the set of integers that cannot be the largest denominator of such a
representation has density zero, and establish its order of growth. Both
results extend to representations of any positive rational number.
