Distinct subset sums in the Conway–Guy sequence

Prove that every set generated by the Conway–Guy sequence has distinct subset sums and determine whether these sets are close to optimal with respect to their largest element.

Background

The Conway–Guy sequence is a construction of sets of integers intended to provide efficient examples with distinct subset sums and thereby upper bounds for the ES-sequence. Conway and Guy verified the property for the first 40 sets and proposed that the construction continues to have distinct subset sums indefinitely.

The conjecture also asserts near-optimality concerning the largest element, which would clarify how closely the construction approaches the minimum possible maximum element in the Erdős subset-sum problem. The paper reports later verification for finite initial ranges but does not state that the conjecture has been completely resolved.

References

They showed that the first 40 sets of the Conway-Guy sequence have distinct subset sums and conjectured that all sets arising from their construction have distinct subset sums and are close to the best possible (with respect to the largest element).

— Erdős Conjecture and AR-Labeling  (2502.19182 - Manattu et al., 26 Feb 2025) in Section 2, “Erdős Subset Sum Conjecture and ES-sequence”