Conway–Guy sequence conjecture

Establish that every set generated by the Conway–Guy construction 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 provide near-optimal examples for the Erdős subset sum problem. The paper states that only the first 40 sets were initially verified to have distinct subset sums, while later computational and theoretical work established the property for broader finite ranges and, separately, proved the distinct-subset-sum property for all sets arising from the construction.

The quoted conjecture concerns both the persistence of distinct subset sums throughout the construction and the near-optimality of the resulting largest elements.

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”