Sum of Squares Rank of Biquadratic Forms and The Zarankiewicz Number

Denote the maximum sos rank of $m\times n$ sum of squares (SOS) biquadratic forms by $BSR(m,n)$. In this paper, we show that $BSR(m,n)\ge z(m,n)$ and conjecture that $BSR(m,n)=z(m,n)$, where $z(m,n)$ is the Zarankiewicz number. Our result coincides with the existing results for $m=2$, $n=2$, and $m=n=3$, and is superior to other previously known lower bounds. Our result also connects graph theory and SOS polynomial theory.