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 < 1 BI <I, and xhi B h = 0 (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
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