Shin’s four-element label-level realization problem

Determine whether every value of |hA| attained by a four-element integer set A is attained by a four-element integer set contained in [0,D_h^{\max}], equivalently whether ν(h,4)=D_h^{\max} for every h≥2.

Background

This problem concerns the label-level invariant |hA|, the number of distinct h-term sums of a four-element integer set, rather than the finer ordered zero-relation profile studied in the paper.

Here D_h{\max} denotes the known extremal diameter quantity L_4(h), and ν(h,4) is the least interval length sufficient to realize every value of |hA| for four-element integer sets. The paper explicitly notes that its sharp ordered-model theorem does not resolve this weaker label-level question.

References

The open problem in nearest to this paper is the one raised after his Corollary~10.5: whether $\nu(h,4)=D_h{\max}$ for every $h\ge2$, that is, whether every value of $|hA|$ over four-element integer sets is attained inside $[0,D_h{\max}]$.

— Sharp order-preserving integer models for short additive equalities  (2609.08915 - Zhang, 8 Sep 2026) in Question M-q:shin-label, Section 8 (Open problems)