A sum packing problem of Erdös and the Conway-Guy sequence

Abstract:

A set $S$ of positive integers has distinct subset sums if the set $\left \{ \sum _{x \in X} x : X \subset S \right \}$ has $2^{|S|}$ distinct elements. Let \[ f(n) = \min \{ \max S: |S|=n \hskip 2mm \text {and} \hskip 2mm S \hskip 2mm \text {has\hskip 2mm distinct\hskip 2mm subset \hskip 2mm sums}\}. \] In 1931 Paul Erdős conjectured that $f(n) \ge c2^{n}$ for some constant $c$. In 1967 John Conway and Richard Guy constructed an interesting sequence of sets of integers. They conjectured that these sets have distinct subset sums and that they are close to the best possible (with respect to largest element). We prove that sets from this sequence have distinct subset sums. We also present some variations of this construction that give microscopic improvements in the best known upper bound on $f(n)$.