Conway–Guy sequence distinct-subset-sums conjecture

Prove that every set generated by the Conway–Guy sequence has distinct subset sums and is close to optimal with respect to its largest element.

Background

The Conway–Guy sequence is a construction of sets of integers intended to produce many elements with distinct subset sums while keeping the largest element small. The paper reports that the first 40 sets in the construction were verified to have distinct subset sums and that the construction was conjectured to remain valid for all sets it generates.

If true, the conjecture would provide strong upper-bound constructions for the ES-sequence, thereby improving the range of available labels for AR-labelings and influencing estimates of the AR-index of graphs.

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”