TITLE: Egyptian representations of $1$
QUESTION [11 upvotes]: Let $\ F(n)\ (\mbox{where}\ n\in\mathbb N:=\{1\ 2\ \ldots\})\ $ be the least cardinality $\ |A|\ $ of a set $\ A\subseteq\mathbb N  $ such that:

$\ \min A=n $
$\ \sum_{x\in A}\frac 1x = 1 $

QUESTION   Is set $\ \{n\in\mathbb N:\ \frac{F(n)}n\le 2\}\ $ finite?

Background:

Roughly speaking, $\ \frac {F(n)}n\ \ge\ e-1,\ $ where $\ e:=exp(1)\ $ is one of the Euler's constants, $\ 2.718\ldots.\ $ This induces a bunch of natural questions, I have restricted myself to just one. Please, feel free to enjoy addressing also related questions.

REPLY [14 votes]: Lots of questions along these lines were raised by Erdos and Graham and many have been solved by Croot, Greg Martin and others.  In particular, Croot has shown that any rational number $r$ can be represented as a 
sum of unit fractions with denominators lying in the interval $[N, (e^r+o(1))N]$ for all large $N$ (indeed his result is more precise, quantifying also the $o(1)$ term).   In particular your $F(N)$ is $\sim (e-1)N$ for all large $N$.  For related work, see also Martin.