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}
}
  • Ernie Croot
  • Published 2003
  • Mathematics
  • Annals of Mathematics
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. 
Ramsey Theory on the Integers
Preliminaries Van der Waerden's theorem Supersets of $AP$ Subsets of $AP$ Other generalizations of $w(k r)$ Arithmetic progressions (mod $m$) Other variations on van der Waerden's theorem Schur'sExpand
The number of representations of rationals as a sum of unit fractions
For given positive integers m and n, we consider the frequency of representations of m n as a sum of unit fractions.
Egyptian Fractions
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 resultsExpand
Old and New Problems and Results in Ramsey Theory
In this note, I will describe a variety of problems from Ramsey theory on which I would like to see progress made. I will also discuss several recent results which do indeed make progress on some ofExpand
An Excursion into Nonlinear Ramsey Theory
TLDR
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
Some of My Favorite Problems in Ramsey Theory
TLDR
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
Paul Erdős and Egyptian Fractions
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 resultsExpand
On Egyptian fractions of length 3
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
Sums of four and more unit fractions and approximate parametrizations
TLDR
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
Paul Erdős and Egyptian Fractions
The Rhind Papyrus of Ahmes [47] (see also [34, 63]) is one of the oldest known mathematical manuscripts, dating from around 1650 B.C. It contains among other things, a list of expansions of fractionsExpand
...
1
2
...

References

SHOWING 1-7 OF 7 REFERENCES
Unsolved Problems in Number Theory
This monograph contains discussions of hundreds of open questions, organized into 185 different topics. They represent aspects of number theory and are organized into six categories: prime numbers,Expand
Ten lectures on the interface between analytic number theory and harmonic analysis
Uniform distribution van der Corput sets Exponential sums I: The methods of Weyl and van der Corput Exponential sums II: Vinogradov's method An introduction to Turan's method Irregularities ofExpand
Sieve Methods
Preface Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of theExpand
Unsolved Problems in Number Theory, Second edition
  • Unsolved Problems in Number Theory, Second edition
  • 1994
Iii
over native-speaking users of English. Secondly, the numerical preponderance of non-native speakers means that it is their communication which is increasing more rapidly and thus dominating theExpand
Old and new problems and results in combinatorial number theory