On a conjecture of Erdős, Graham and Spencer

It is conjectured by Erdős, Graham and Spencer that if 1⩽a1⩽a2⩽⋯⩽as with , then this sum can be decomposed into n parts so that all partial sums are ⩽1. This is not true for as shown by a1=2, a2=a3=3, a4=⋯=a5n−3=5. In 1997, Sándor proved that Erdős–Graham–Spencer conjecture is true for . In this paper, we reduce Erdős–Graham–Spencer conjecture to finite calculations and prove that Erdős–Graham–Spencer conjecture is true for . Furthermore, it is proved that Erdős–Graham–Spencer conjecture is true if and no partial sum (certainly not a single term) is the inverse of an positive integer.