On a coloring conjecture about unit fractions (Q1425317)

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 \textit{R. L. Graham} and \textit{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. \textit{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 \textit{T. C. Brown} and \textit{V. Rödl} [Bull. Aust. Math. Soc. 43, No. 3, 387--392 (1991; Zbl 0728.11017)] and \textit{S. D. Adhikari} and \textit{R. Thangadurai} [Monochromatic solutions of Diophantine equations, in Algebraic number theory and Diophantine analysis (Graz, 1998), 1--9 (2000; Zbl 0958.11027)].