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On a coloring conjecture about unit fractions. (English) Zbl 1041.11019

The author proves the following beautiful result: Let \(b=e^{167000}\). If one colours the integers in \([2,b^r]\) with \(r\) colours, then there exists a monochromatic set \(S\in [2,b^r]\) such that \(\sum_{n \in S} \frac{1}{n}=1\).
This answers a problem, posed by R. L. Graham and P. Erdős [Old and new problems and results in combinatorial number theory, Enseign. Math. Monogr. No. 28, Genève (1980; Zbl 0434.10001)]. Graham has rewarded this solution with a $500-cheque, blanksigned by Erdős.
The proof is relatively self-contained using methods from multiplicative number theory. For similar techniques and results see also the author’s work [Acta Arith. 99, No. 2, 99–114 (2001; Zbl 0979.11022) and Mathematika 46, No. 2, 359–372 (1999; Zbl 1033.11014)].
It is clear that \(b< e\) would not suffice. Andreas Hipler [Zu Modulspannweiten und zu einem Problem von Erdős und Graham (Ph.D. Thesis, Mainz, 2002)] proved that for \(r=2\) the smallest value guaranteeing a monochromatic solution is 208.
Earlier work on monochromatic solutions had been done on homogeneous equations such as \(\frac{1}{x}+ \frac{1}{y}= \frac{1}{z}\). For this see T. C. Brown and V. Rödl [Bull. Aust. Math. Soc. 43, No. 3, 387–392 (1991; Zbl 0728.11017)] and S. D. Adhikari and R. Thangadurai [Monochromatic solutions of Diophantine equations, in Algebraic number theory and Diophantine analysis (Graz, 1998), 1–9 (2000; Zbl 0958.11027)].

MSC:

11D68 Rational numbers as sums of fractions
11B75 Other combinatorial number theory
05D10 Ramsey theory
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