0-partition number vs partition number Denote by $\chi_0(f)$ the minimum number of 0 monochromatic rectangles needed to cover the 0-inputs of $f$, and by $\chi(f)$ the minimum number of monochromatic rectangles needed to cover the all inputs of $f$ (both 0 and 1). It is known that

$$\log \chi(f) \leq D(f) \leq (\log \chi_0(f))^2.$$

The last inequality follows from the famous clique vs independent set upper bound. Is there an $f$ that witnesses such a quadratic separation between $\log \chi(f)$ and $\log \chi_0(f)$?
A quadratic separation between $\log\chi_0$ and $\log\chi_1$ is proved in a paper by Göös, Jayram, Pitassi and Watson, see the ECCC report.

In Theorem 2 they construct a function $F$ with small $\log\chi_1(F)$, and they say "In fact, we prove Theorem 2 by showing that (the negation of) the function $F$ has high approximate nonnegative rank". Observe that $\log\chi_0(F)$ is at least the nonnegative rank of the negation of $F$.