Erdős subset-sum growth conjecture

Prove that for every finite set of positive integers with distinct subset sums, whose largest element is a_n, there exists an absolute constant c>0 such that a_n>c\cdot 2^n.

Background

The paper studies finite sets of positive integers whose 2^n subset sums are all distinct and defines ES(n) as the smallest possible largest element of an n-element set with this property. The Erdős subset-sum conjecture predicts an exponential lower bound on this largest element, namely a bound proportional to 2^n.

The paper notes that the best known general lower bound remains of the form ES(n)>c\cdot 2^n/\sqrt{n}, so establishing the conjectured order of growth would require eliminating the factor 1/\sqrt{n}. This conjecture is used to motivate the ES-sequence and its applications to lower bounds for the AR-index of graphs.