On a conjecture of Erdös, Graham and Spencer, II

It is conjectured by Erdős, Graham and Spencer that if 1 ≤ a1 ≤ a2 ≤ · · · ≤ as are integers with ∑s i=1 1/ai < n − 1/30, then this sum can be decomposed into n parts so that all partial sums are ≤ 1. This is not true for ∑s i=1 1/ai = n − 1/30 as shown by a1 = · · · = an−2 = 1, an−1 = 2, an = an+1 = 3, an+2 = · · · = an+5 = 5. In 1997 Sandor proved that Erdős–Graham–Spencer conjecture is true for ∑s i=1 1/ai ≤ n − 1/2. Recently, Chen proved that the conjecture is true for ∑s i=1 1/ai ≤ n − 1/3. In this paper, we prove that Erdős–Graham–Spencer conjecture is true for ∑s i=1 1/ai ≤ n − 2/7. c © 2008 Elsevier B.V. All rights reserved.