Zarankiewicz numbers near the triple system threshold

For positive integers m $m$ and n $n$, the Zarankiewicz number Z2,2(m,n) ${Z}_{2,2}(m,n)$ can be defined as the maximum total degree of a linear hypergraph with m $m$ vertices and n $n$ edges. Guy determined Z2,2(m,n) ${Z}_{2,2}(m,n)$ for all n >= m2/3+O(m) $n\geqslant \left(\genfrac{}{}{0.0pt}{}{m}{2}\right)\unicode{x02215}3+O(m)$. Here, we extend this by determining Z2,2(m,n) ${Z}_{2,2}(m,n)$ for all n >= m2/3 $n\geqslant \left(\genfrac{}{}{0.0pt}{}{m}{2}\right)\unicode{x02215}3$ and, when m $m$ is large, for all n >= m2/6+O(m) $n\geqslant \left(\genfrac{}{}{0.0pt}{}{m}{2}\right)\unicode{x02215}6+O(m)$.