Counting Sets of Integers with Various Summation Properties

Abstract

The motivation behind these investigations comes form a problem due to P.J. Cameron and P. Erdös, presented at the Fourteenth British Combinatorial Conference (1993) [1, 2]. They conjectured that

$$ \frac{{S\left( s \right)}}{{{2^{n/2}}}} \to {C_0} or {C_E} $$

(for constants C ₀ and C _E ), as n→∞ through odd or even values respectively, where S(n) is the number of sum-free subsets of the first n natural numbers (i.e. containing no solution to x + y = z, where x and y are not necessarily distinct).