Overtake and the recursive-line equality for second-order Zarankiewicz numbers

Abstract

We study two questions about the second-order Zarankiewicz numbers. The first is the recursive-line equality $z_2(m,n)=z_{RL}(m,n)$, known to hold in every computed case and conjectured in general. The second, here called \emph{overtake}, is the strict inequality $z_{2A}(m,n)>z_2(m,n)$ between the second-order augmented number and the second-order number. Both are governed by the number $H_{\mathrm{limited}}(m,n)$ of unoccupied cells in a limited extremal configuration. We prove that the first increase $z_2\to z_2+1$ already costs three unoccupied cells, forcing a unit of one-edge deficit, and the cell identity then bounds the excess $z_{2A}-z_2$ by $\lfloor (H_{\mathrm{limited}}-1)/2\rfloor$. In particular, when the recursive-line hole count is at most two, no overtake occurs. We conjecture the sharp budget $\lfloor H_{\mathrm{limited}}/3\rfloor$ and that $H_{\mathrm{limited}}\ge 5$ is the threshold for an overtake. In the computed range $n\le 7$, no overtake occurs at any order, the sole exception being the known case $(4,4)$. The recursive-line equality $z_2=z_{RL}$ 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.