Of a Set of Integers
نویسنده
چکیده
For r-~2 let p(n, r ) denote the maximum cardinality of a subset A of N = { 1 , 2 . . . . , n} such that there are no B c A and an integer y with S b = y ' . I t is shown that for any e >-0 and bEB n>-n(e), (l+o(l))2~/t'+l>n('-l>/t'+l)~_p(n, r)~_n~§ for all r_~5, and that for every fixed r~_6, p(n,r)=(l+o(1)).21/t'+~)n (~-1)/('§ as n ~ . Let f(n,m) denote t h e m a x i m u m c a r d i n a l i t y o f a subset A of N such that there is no BoA the sum of whose dements is m. I t is proved that for 3n6JS*'~-m~--n~/20 log ~ n and n~-n(6), f(n, m)=[n[sJ+s-2, where s is the smallest integer that does not divide m. A special case of this result establishes a conjecture of Erd6s and Graham.
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