Linear realizations for small k

Establish that, whenever k<y/2, every multiset of the form {1^a,(y−k)^b,y^c} has a linear realization for all a≥y.

Background

The paper develops standard linear realizations for multisets with support {1,y−k,y}. It proves the asserted lower bound a≥y for k≤2, with additional constructions for k≤4 under divisibility or small-k conditions.

Conjecture 1.11 proposes the natural extension to all k<y/2. Resolving it would substantially extend the paper’s construction methods and provide a significant step toward the Coprime BHR Conjecture for supports of size three.

References

We believe the next logical extension of our methods could provide a resolution to the following conjecture, which would be a significant step towards a proof of the Coprime BHR Conjecture for supports of size 3. Conjecture 1.11. (Linear realizations for small k) When k < y/2, multisets of the form {1a, (y − k)b, yc} are linearly realizable for all a ≥ y.

— Construction Techniques for Linear Realizations of Multisets with Small Support  (2502.00164 - Ağırseven et al., 31 Jan 2025) in Conjecture 1.11, Section 1 (Introduction)