Second Order Zarankiewicz Number

Abstract

We introduce the \emph{second order Zarankiewicz number} $z_2(m,n)$ for irreducible doubly simple biquadratic forms with $|E_1|=z(m,n)$, together with the intermediate recursive-line and signed parameters $z_{RL}(m,n)$ and $z_{SL}(m,n)$. They satisfy the unconditional hierarchy \[ \operatorname{BSR}(m,n) \ge z_2(m,n) \ge z_{SL}(m,n) \ge z_{RL}(m,n) \ge z_{wL}(m,n) \ge z(m,n), \] where $z_{RL}$ is defined by the strengthened recursive rectangle criterion $(RW{3}^{+})$ together with the conditions $(S)$ and $C_4$-freeness of $G_1$, and $z_{SL}$ is defined by the signed criterion $(RW{3}^{\pm })$. We show that $(RW{3}^{+})$ is sound and strictly weaker than the literal weak cross-cell test $(W3)$ on the weak-admissible class, and that $(RW{3}^{\pm })$ is likewise sound. At the smallest four-column cases we obtain \[ z_2(5,4)=z_{SL}(5,4)=z_{RL}(5,4)=13>12=z_{wL}(5,4), \] and \[ z_2(6,4)=z_{SL}(6,4)=z_{RL}(6,4)=16>14=z_{wL}(6,4). \] In three columns this yields \[ z_2(m,3)=z_{RL}(m,3)=2m \qquad \text{for all } m\ge 3, \] with strict separation from $z_{wL}(m,3)$ for every $m\ge 10$, and exact gap $\lfloor (m-3)/3\rfloor$ for $m\ge 16$. Further finite computations give $z_{RL}(5,5)=17$, $z_{RL}(7,4)=19$, and $z_{RL}(7,7)\ge 32>28=z_{wL}(7,7)$. Along $N=2p$ with $p$ an odd prime, we obtain the cubic asymptotic separation \[ z_2\!\left(\binom{N}{2},N\right)-z_{wL}\!\left(\binom{N}{2},N\right)\ge \left(\frac{1}{16}-o(1)\right)N^3. \] For the exceptional incidence case $p=3$ we prove $z_{SL}(15,6)=z_2(15,6)=60$. Concurrent work of Chen and Chen, using the recursive-line definition introduced here, incorporates the finite recursive-line values of this manuscript and establishes further exact four-column values together with an eventual formula for all $m\ge 15$. The conjectural equality $z_2=z_{RL}$ is supported by the exact two-column, three-column, four-column, and odd-prime incidence families.