  Dvo\v{r}\'ak and Kawarabayashi [European Journal of Combinatorics, 2017]
asked, what is the largest chromatic number attainable by a graph of treewidth
$t$ with no $K_r$ subgraph? In this paper, we consider the fractional version
of this question. We prove that if $G$ has treewidth $t$ and clique number $2
\leq \omega \leq t$, then $\chi_f(G) \leq t + \frac{\omega - 1}{t}$, and we
show that this bound is tight for $\omega = t$. We also show that for each
value $0 < c < \frac{1}{2}$, there exists a graph $G$ of a large treewidth $t$
and clique number $\omega = \lfloor (1 - c)t \rfloor$ satisfying $\chi_f(G)
\geq t + 1 + \frac{1}{2}\log(1-2c) + o(1)$, which is approximately equal to the
upper bound for small values $c$.
