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

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 \(6\le m\le 16\).