Exact relation-free diameter for five-element alphabets

Determine L_5(q), the least diameter of a five-element integer alphabet having no nonzero balanced relation of mass at most q, for arbitrary q≥2.

Background

The quantity L_5(q) is the minimum diameter of a relation-free five-element integer alphabet and, because the relation-free profile is unique, also equals the maximum minimum model diameter among relation-free five-letter alphabets.

The paper proves only the general upper bound L_5(q)≤q2(q+1). Exhaustive searches determine L_5(2)=11, L_5(3)=23, and L_5(4)=41, but the explicit general construction gives larger values for q=3 and q=4, showing that the construction is not generally optimal.

References

The present paper does not determine $L_5(q)$ for arbitrary $q$; exhaustive search gives $L_5(2)=11$, attained by ${0,1,4,9,11}$ (Corollary~\ref{M-cor:six-explanations}), $L_5(3)=23$, attained by ${0,1,15,18,23}$, and $L_5(4)=41$, attained by ${0,1,24,37,41}$, whereas the construction of the corollary gives $11$, $34$ and $61$; so that construction is not optimal in the two cases $q=3,4$.

— Sharp order-preserving integer models for short additive equalities  (2609.08915 - Zhang, 8 Sep 2026) in Question M-q:L5, Section 8 (Open problems); related discussion after Corollary M-cor:five-rank-zero