On a conjecture of Erd?s, Graham and Spencer, II
It is conjectured by Erd?s, Graham and Spencer that if 1?a1?a2???as 1 ? a 1 ? a 2 ? ? ? a s are integers with ? i = 1 s 1 / a i < n ? 1 / 30 , then this sum can be decomposed into n n parts so that all partial sums are ?1 ? 1 . This is not true for ? i = 1 s 1 / a i = n ? 1 / 30 as shown by a1=?=an?2=1 a 1 = ? = a n ? 2 = 1 , a n ? 1 = 2 , a n = a n + 1 = 3 , a n + 2 = ? = a n + 5 = 5 . In 1997 Sandor proved that Erd?sGrahamSpencer conjecture is true for ? i = 1 s 1 / a i ? n ? 1 / 2 . Recently, Chen proved that the conjecture is true for ? i = 1 s 1 / a i ? n ? 1 / 3 . In this paper, we prove that Erd?sGrahamSpencer conjecture is true for ? i = 1 s 1 / a i ? n ? 2 / 7 .
erd?sgrahamspencer conjecture partition