On the Erdős–Graham–Spencer conjecture
Volume 151 / 2018
Colloquium Mathematicum 151 (2018), 203-215
MSC: Primary 11B75.
DOI: 10.4064/cm6946-2-2017
Published online: 13 December 2017
Abstract
In 1980, Erdős, Graham and Spencer conjectured that if $1\leq a_1\leq \cdots \leq a_s$ are integers with $\sum _{i=1}^s 1/a_i \lt n-1/30$, then this sum can be decomposed into $n$ parts so that all partial sums are $\leq 1$. In 1997, Sándor proved that the Erdős–Graham–Spencer conjecture is true for $\sum _{i=1}^s 1/a_i \leq n-1/2$. In 2006, Chen improved $1/2$ to $1/3$. Afterwards, Fang and Chen improved $1/3$ to $2/7$. In this paper, we prove that the Erdős–Graham–Spencer conjecture is also true for $\sum _{i=1}^s 1/a_i \leq n-3/11$.
