Density version of the Erdős-Graham conjecture

Question

In 2003 E. S. Croot [Ann. of Math. 157(2)(2003), 545-556] proved the Erdős-Graham Conjecture which states that if $\{2,3,\ldots\}$ is partitioned into finitely many subsets then one of the subsets contains finitely many distinct integers $x_1,\ldots,x_m$ satisfying $\sum_{k=1}^m1/x_k=1$.

Here I ask for the density version of this result.

QUESTION: Let $A$ be a subset of $\{2,3,\ldots\}$ having positive lower (or upper) asymptotic density. Are there finitely many distinct elements $a_1 <\ldots< a_m$ of $A$ with $\sum_{k=1}^m1/a_k = 1$?

Clearly, the set $\{3,5,7,\ldots\}$ has asymptotic density $1/2$, and it is known that $$\frac1 3 + \frac1 5 +\frac 1 7 +\frac 1 9 +\frac 1 {11} +\frac 1 {33} + \frac1{35} + \frac1 {45} + \frac 1 {55} + \frac1 {77} + \frac1 {105} = 1.$$

Motivated by Szemerédi's theorem, in 2007 I formulated the above question and conjectured that it has a positive answer. It seems that Croot's proof of the Erdős-Graham conjecture could not be adapted to answer my question.

Any comments are welcome!

Answer 1

The answer is yes (even to the strong upper density version), as proved in this preprint https://arxiv.org/abs/2112.03726.

(I can give this answer with a high degree of confidence, since the proof is now completely formalised in Lean! This was a joint project by myself and Bhavik Mehta - see https://github.com/b-mehta/unit-fractions for the code)

Answer 2

Naturally, this was also considered by Erdös and Graham. Graham mentions at the top of page 10 here for instance: http://www.math.ucsd.edu/~ronspubs/13_03_Egyptian.pdf. I'm not aware of any progress.