Relationship between second-order and recursive-line Zarankiewicz numbers

Determine the relationship between the second-order Zarankiewicz number $z_2(m,n)$ and the recursive-line Zarankiewicz number outside the parameter families where both quantities attain a common upper bound.

Background

The paper establishes a hierarchy relating the biquadratic SOS rank, the second-order number z2z_2, the recursive-line number, and the classical Zarankiewicz number. For the four-column families constructed in the paper, z2(m,4)z_2(m,4) and the recursive-line number coincide because both attain the same cell-count upper bound.

Beyond such common-upper-bound cases, the recursive closure criterion may fail even when the associated SOS representation remains irreducible. Consequently, the paper does not settle whether the two parameters coincide or differ in general.

References

Finally, the relationship between $z_2$ and $$ remains unresolved outside the families in which both meet a common upper bound.

— Recursive-Line Zarankiewicz Numbers with Four Columns  (2609.11093 - Chen et al., 10 Sep 2026) in Section 5.2, final paragraph