Is the Signed Zarankiewicz Number the Same as the Recursive-line Zarankiewicz Number?

Source abstract

Löfberg and Qi introduced the second order Zarankiewicz number $z_2$, the recursive-line Zarankiewicz number $z_{RL}$, and the signed Zarankiewicz number $z_{SL}$ for doubly simple biquadratic forms. It was shown that $$z_2(m,n)\ge z_{SL}(m,n)\ge z_{RL}(m,n)$$ for all $m$ and $n$. However, there was no evidence that there exist particular $m$ and $n$ such that $z_{SL}(m,n)>z_{RL}(m,n)$. The motivation for introducing $z_{SL}$ was as follows: during the study of the exceptional case $m=15$, $n=6$, Löfberg and Qi showed that $$z_2(15,6)=z_{SL}(15,6)=60,$$ but the exact value of $z_{RL}(15,6)$ was unknown then. In this paper we show that $$z_{RL}(15,6)=60.$$ This eliminates the motivation for introducing $z_{SL}$. Whether $z_{SL}(m,n)=z_{RL}(m,n)$ in general remains an open problem. Recently, Lebedev presented an explicit construction separating the augmented Zarankiewicz number $z_A$ from the limited augmented Zarankiewicz number $z_L$ at $m=n=1893$. We hope that the separation problem for $z_{SL}$ and $z_{RL}$ can also be solved. We also present the exact values of $z_{RL}(m,6)$ for $6\le m\le 16$.