Existence beyond the quadratic row threshold

Establish whether $(r \times c, v)$-near triple arrays exist for every c at least r(r-1) and every v at least c.

Background

The paper proves existence for all (3×c,v)(3 \times c,v)-near triple arrays with c at least 6 and v at least c, while proving non-existence for a family with c=r(r-1)-1 when r is at least 4. These results motivate the proposed threshold conjecture that c at least r(r-1) suffices for existence for all admissible v at least c.

References

We therefore propose the following conjecture. $(r \times c, v)$-near triple arrays exist for any $c \geq r(r - 1)$ and $v \geq c$.

— Near Triple Arrays  (2503.07166 - Gordeev et al., 10 Mar 2025) in Conjecture in Section 7, Concluding remarks