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Summary: JOURNAL OF NUMBER THEORY 27. 196-205 (1987)
Subset Sums
N. ALON*
Bell Communications Research. 435 South Street Morristown.
New' Jersey 07960, and Depurtment of Mathemarics.
Tel .4oitr University. Tel Aoio, Israel
C'owwtumc~u~ed hy R. L. Gruhum
Received September 8. 19X6
Suppose E >O and k > I. We show that if II > n,,(k. a) and .4 L Z,, satisfies
IAl > (( l/k) + E)n then there is a subset B L A such that 0 < 1BI
(in 2,). The case k = 3 solves a problem of Stalley and another problem of Erdos
and Graham. For an integer HI > 0, let snd(nt) denote the smallest integer that does
not divide PI. We prove that for every I-:> 0 there is a constant c = ~(8:) z I, such
that for every n > 0 and every rn, n ' +' 6 WI < n'llog'n every set A E j I, Z,..., II ) of
cardinality IAl > c.n/snd(m) contains a subset Bcl- .4 so that ChcB h=m. This is
best possible, up to the constant C. In particular it implies that for every II there is
an m such that every set A c (l,..., II jof cardinality IAl > cx/log II contains a subset
BG A so that xhtS h = ,n, thus settling a problem of Erdds and Graham. ' IYX7
Academic Press. Inc
1. l~TR00uCTl0N
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