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!


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.


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