 
Summary: COMBINAIORICA
Akad6miai Kiad6 ~ SpringerVerlag
ON SUMS OF SUBSETS
COM~INATOFaCA8 (4) 1988297306
OF A SET OF INTEGERS
N. ALON* and G. FREIMAN
ReceivedJuly 1, 1987
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 Bc A and an integer y with S b=y'. It 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 themaximumcardinalityof
a subset A of N such that there is no BoA the sum of whose dements is m. It is proved that for
3n6JS*'~m~n~/20log~n and n~n(6), f(n, m)=[n[sJ+s2, where s is the smallest integer that
does not divide m. A special case of this result establishes a conjecture of Erd6s and Graham.
Introduction
Let n be an integer and define N= {1, 2..... n}. For a set AcN, let A*
denote the set of all sums of subsets of A, i.e. A*={b~ b: BC=A}. There are several
recent and less recent problems and results, that assert that if IAI is large enough,
then A* must contain some numbers of prescribed type. See [5], [3], [I], [2], [4]. In
