ON THE STRUCTURE AND THE NUMBER OF SUM-FREE SETS

A finite set A of Positive integers is called sum-free if A and (A + A) = 0. For n odd, {1, 3, 5,..., n} and {n + 1/2, n + 3/2,..., n} are examples of such sets. Denote by m and l, respectively, the largest and smallest elements of A and by a the cardinality of A. We show that if the cardinality of the sum-free set A does not differ much from l/2, then A does not differ much from one of the two examples mentioned above. More precisely, if a > 5/12l + 2, then either all elements of A are odd or A contains both odd and even integers and m greater-than-or-equal-to a. It is shown that if a > 5/12l + 2 then the number of such sum-free sets is O(2n/2) which proves for such sets the conjecture of P. Cameron and P Erdos.