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

Abstract

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 $m=\left(\genfrac{}{}{0ex}{}{4t}{2}\right)$ and $n=4t$. Then $$\mathrm{BSR}(m,n)\ge z_L(m,n)\ge 2\left(\genfrac{}{}{0ex}{}{4t}{2}\right)+4t^2-2t.$$ Since $z=2\left(\genfrac{}{}{0ex}{}{4t}{2}\right)=\hat{I}˜(t^2)$, the gap satisfies $z_L-z\ge 4t^2-2t=\hat{I}˜(t^2)=\hat{I}˜(m)$, i.e., it grows linearly in $m$. Moreover, $$\frac{z_L-z}{z}\ge \frac{4t^2}{16t^2-4t}⟶\frac{1}{4}\text{as }t\to \infty ,$$ so the gap is asymptotically at least $25\%$ 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 $5\times 3$ and $5\times 4$: $z_L(5,3)=9$ and $z_L(5,4)=12$. 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: $z_L(6,3)\ge 10$ and $z_L(6,5)\ge 17$. For $5\times 5$ we establish a new lower bound $z_L(5,5)\ge 15$, improving the previously known bound. By a direct construction, we have $z_L(6,6)\ge 20$.