In 2003 E. S. Croot [Ann. of Math. 157(2)(2003), 545-556] proved the Erdos-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 Szemeredi'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 Erdos-Graham conjecture could not be adapted to answer my question.


Any comments are welcome!



Density version of the Erdos-Graham conjecture
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)


