A General Lower Bound for the Limited Augmented Zarankiewicz Number based upon Complete Graphs

The limited augmented Zarankiewicz number $z_L(m,n)$ satisfies $\mathrm{BSR}(m,n)\ge z_L(m,n)\ge z(m,n)$, where $\mathrm{BSR}(m,n)$ is the maximum SOS rank of $m\times n$ biquadratic forms and $z(m,n)$ is the classical Zarankiewicz number. Our main result is a general lower bound for $z_L(m,n)$ based on the incidence graph of the complete graph $K_{4t}$. For every integer $t\ge 1$, let $m=\left(\genfrac{}{}{0px}{}{4t}{2}\right)$ and $n=4t$. Then $$z_L(m,n)\ge 2\left(\genfrac{}{}{0px}{}{4t}{2}\right)+4t^2-2t\mathrm{.}$$ Consequently, $$\mathrm{BSR}(m,n)\ge 2\left(\genfrac{}{}{0px}{}{4t}{2}\right)+4t^2-2t\mathrm{.}$$ Since $z=2\left(\genfrac{}{}{0px}{}{4t}{2}\right)=\mathrm{\Theta }(t^2)$, the gap satisfies $z_L-z\ge 4t^2-2t=\mathrm{\Theta }(t^2)=\mathrm{\Theta }(m)$, i.e., it grows linearly in $m$. Moreover, $$\frac{z_L-z}{z}\ge \frac{4t^2}{16t^2-4t}⟶\frac{1}{4}\text{as }t\to \mathrm{\infty },$$ so the gap is asymptotically at least $25\mathrm{\%}$ of $z$ – a non-negligible constant fraction. For $t=1$ we obtain $z_L(6,4)\ge 14$, and we prove that this bound is tight, i.e., $z_L(6,4)=14$. For $t=2$ and $t=3$ we obtain $z_L(28,8)\ge 68$ and $z_L(66,12)\ge 162$, respectively, improving previously known bounds. We also determine the exact values of $z_L(m,n)$ for all $m,n\le 5$: $z_L(5,3)=9$, $z_L(5,4)=12$, and $z_L(5,5)=14$, confirming that previously known lower bounds are tight. These results serve as base cases for a \emph{lifting method} that constructs admissible limited augmented graphs on $(m+1)\times (n+1)$ from optimal ones on $m\times n$. Applying this method, we obtain new lower bounds: $$z_L(6,3)\ge 10,z_L(6,5)\ge 17,z_L(6,6)\ge 19,$$ where $z(6,3)=9$, $z(6,5)=14$, and $z(6,6)=16$.