A Counterexample to a Conjecture of Erdős, Graham and Spencer

Abstract

It is conjectured by Erdős, Graham and Spencer that if $1 \leq a_1 \leq a_2 \leq \cdots \leq a_s$ with $\sum_{i=1}^s 1/a_i < n - 1/30$, then this sum can be decomposed into $n$ parts so that all partial sums are $\leq 1$. In this note we propose a counterexample which gives a negative answer to this conjecture.