On a conjecture of Erdős, Graham and Spencer. II (Q1005237)

It is conjectured by Erdős, Graham and Spencer that if \(a_1 \leq a_2 \leq \ldots \leq a_s\) are integers 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 \(\leq 1\). This is not true for \(\sum_{i=1}^s 1/a_i = n-1/30\) as shown by \(a_1=\dots=a_{n-2}=1\), \(a_{n-1} = 2\), \(a_n = a_{n+1} = 3\) , \(a_{n+2} = \dots = a_{n+5} = 5\). In 1997 \textit{C. Sándor} [J. Number Theory 63, No. 2, 203--210 (1997; Zbl 0876.11006)] proved that Erdős-Graham-Spencer conjecture is true for \(\sum_{i=1}^s 1/a_i \leq n-1/2\). Recently, \textit{Y.-G. Chen} [Part I, J. Number Theory 119, No. 2, 307--314 (2006; Zbl 1183.05007)] proved that the conjecture is true for \(\sum_{i=1}^s 1/a_i \leq n-1/3\). In this paper, we prove that Erdős-Graham-Spencer conjecture is true for \(\sum_{i=1}^s 1/a_i \leq n-2/7\).