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On the Erdős–Graham–Spencer conjecture

Volume 151 / 2018

Meng-Long Yu, Jin-Hui Fang 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 $\sum _{i=1}^s 1/a_i \leq n-3/11$.

Authors

  • Meng-Long YuDepartment of Mathematics
    Nanjing University of Information Science & Technology
    210044 Nanjing, P.R. China
    e-mail
  • Jin-Hui FangDepartment of Mathematics
    Nanjing University of Information Science & Technology
    210044 Nanjing, P.R. China
    e-mail

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