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

Do we have

z_2(6,6)=z_{SL}(6,6)=z_{RL}(6,6) \quad\text{and}\quad z_2(7,6)=z_{SL}(7,6)=z_{RL}(7,6)?

— Is the Signed Zarankiewicz Number the Same as the Recursive-line Zarankiewicz Number?  (2609.20071 - Chen et al., 17 Sep 2026) in Section 5, item 3, “Open Problems”

Are there further refinements of the Zarankiewicz number that are strictly stronger than $z_{RL}$ and strictly weaker than $z_2$?

— Is the Signed Zarankiewicz Number the Same as the Recursive-line Zarankiewicz Number?  (2609.20071 - Chen et al., 17 Sep 2026) in Section 5, item 6, “Open Problems”

For every integer n\ge6 with m=\binom n2, z_2\left(\binom n2,n\right)=z_{RL}\left(\binom n2,n\right)=Z(n),\ &\text{that is}\quad z_2(m,n)=z_{RL}(m,n)=\left\lfloor\frac{mn+z(m,n)}2\right\rfloor .

— Exact Second-Order Zarankiewicz Numbers for Complete-Graph Incidence Families  (2609.25974 - Chen et al., 22 Sep 2026) in Conjecture 1, Section 4 (labelled Conjecture~\ref{conj:all-n})

Are there other families where the cell bound is attained and (\mathrm{RW}3+) suffices?

— Exact Second-Order Zarankiewicz Numbers for Complete-Graph Incidence Families  (2609.25974 - Chen et al., 22 Sep 2026) in Section 5, Open problems, item 7

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

The identity zRL = zSL = z2 is established in many cases [10, 4, 5, 3, 9, 15], and it is conjectured to hold for every m, n ≥ 2.

— Overtake and the recursive-line equality for second-order Zarankiewicz numbers  (2610.22875 - Ouyang et al., 16 Oct 2026) in Section 1, The recursive-line equality

The recursive-line equality z2 = zRL is verified at every order in the range where both values are determined; the orders at which it remains unsettled are (12, 4) and (14, 4) at width four, (11, 5) at width five, and (8, 7) at width seven.

— Overtake and the recursive-line equality for second-order Zarankiewicz numbers  (2610.22875 - Ouyang et al., 16 Oct 2026) in Abstract; Section 7, “The four open orders”