On a conjecture of Erdős, Graham and Spencer Original Research Article

It is conjectured by Erdős, Graham and Spencer that if 1less-than-or-equals, slanta1less-than-or-equals, slanta2less-than-or-equals, slantcdots, three dots, centeredless-than-or-equals, slantas with image, then this sum can be decomposed into n parts so that all partial sums are less-than-or-equals, slant1. This is not true for image as shown by a1=2, a2=a3=3, a4=cdots, three dots, centered=a5n−3=5. In 1997, Sándor proved that Erdős–Graham–Spencer conjecture is true for image. In this paper, we reduce Erdős–Graham–Spencer conjecture to finite calculations and prove that Erdős–Graham–Spencer conjecture is true for image. Furthermore, it is proved that Erdős–Graham–Spencer conjecture is true if image and no partial sum (certainly not a single term) is the inverse of an positive integer.