  Given a family $\mathcal G$ of graphs spanning a common vertex $V$, a
cooperative coloring of $\mathcal G$ is a collection of one independent set
from each graph of $\mathcal G$ such that the union of these independent sets
equals $V$. We prove that when $d$ is large, there exists a family $\mathcal G$
of $(1+o(1)) \frac{\log d}{\log \log d}$ forests of maximum degree $d$ that
admits no cooperative coloring, which significantly improves a result of
Aharoni, Berger, Chudnovsky, Havet, and Jiang (Electronic Journal of
Combinatorics, 2020). Our family $\mathcal G$ consists entirely of star
forests, and we show that this value for $|\mathcal G|$ is asymptotically best
possible in the case that $\mathcal G$ is a family of star forests.
