# On a coloring conjecture about unit fractions

@article{Croot2003OnAC, title={On a coloring conjecture about unit fractions}, author={Ernie Croot}, journal={Annals of Mathematics}, year={2003}, volume={157}, pages={545-556} }

We prove an old conjecture of Erdos and Graham on sums of unit fractions: There exists a constant b > 0 such that if we r-color the integers in [2, br]: then there exists a monochromatic set S such that neS 1/n-1.

#### 14 Citations

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For given positive integers m and n, we consider the frequency of representations of m n as a sum of unit fractions.

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One of Paul Erdős’ earliest mathematical interests was the study of so-called Egyptian fractions, that is, finite sums of distinct fractions having numerator 1. In this note we survey various results… Expand

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This paper establishes formulas for the two-color Rado numbers for three families of equations: x + yn = z, x + y2 + c = z; and x + y2 = az, where c and a are positive integers. Expand

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In this brief note, a variety of problems from Ramsey theory on which I would like to see progress made are described and I am offering modest rewards for most of these problems. Expand

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One of Paul Erdős’ earliest mathematical interests was the study of so-called Egyptian fractions, that is, finite sums of distinct fractions having numerator 1. In this note we survey various results… Expand

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Let a, n be positive integers that are relatively prime. We say that a/n can be represented as an Egyptian fraction of length k if there exist positive integers m1, . . . ,mk such that a n = 1 m1 + ·… Expand

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New upper bounds on the number of representations of rational numbers mn as a sum of four unit fractions are proved, giving five different regions, depending on the size of m in terms of n, according to the key point to enable computer programmes to filter through a large number of equations and inequalities. Expand

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