In combinatorial number theory, the Erdős–Graham problem is the problem of proving that, if the set
of integers greater than one is partitioned into finitely many subsets, then one of the subsets can be used to form an Egyptian fraction representation of unity.
-coloring of the integers greater than one, there is a finite monochromatic subset
of these integers such that In more detail, Paul Erdős and Ronald Graham conjectured that, for sufficiently large
, the largest member of
must be at least Euler's constant
[1] Ernie Croot proved the conjecture as part of his Ph.D thesis,[2] and later (while a post-doctoral researcher at UC Berkeley) published the proof in the Annals of Mathematics.
Croot's result follows as a corollary of a more general theorem stating the existence of Egyptian fraction representations of unity for sets
of smooth numbers in intervals of the form
contains sufficiently many numbers so that the sum of their reciprocals is at least six.
The Erdős–Graham conjecture follows from this result by showing that one can find an interval of this form in which the sum of the reciprocals of all smooth numbers is at least
satisfying the conditions of Croot's theorem.
A stronger form of the result, that any set of integers with positive upper density includes the denominators of an Egyptian fraction representation of one, was announced in 2021 by Thomas Bloom, a postdoctoral researcher at the University of Oxford.