  Recently, Dvo\v{r}\'ak, Norin, and Postle introduced flexibility as an
extension of list coloring on graphs [JGT 19']. In this new setting, each
vertex $v$ in some subset of $V(G)$ has a request for a certain color $r(v)$ in
its list of colors $L(v)$. The goal is to find an $L$ coloring satisfying many,
but not necessarily all, of the requests.
  The main studied question is whether there exists a universal constant
$\epsilon >0$ such that any graph $G$ in some graph class $\mathcal{C}$
satisfies at least $\epsilon$ proportion of the requests. More formally, for $k
> 0$ the goal is to prove that for any graph $G \in \mathcal{C}$ on vertex set
$V$, with any list assignment $L$ of size $k$ for each vertex, and for every $R
\subseteq V$ and a request vector $(r(v): v\in R, ~r(v) \in L(v))$, there
exists an $L$-coloring of $G$ satisfying at least $\epsilon|R|$ requests. If
this is true, then $\mathcal{C}$ is called $\epsilon$-flexible for lists of
size $k$.
  Choi et al. [arXiv 20'] introduced the notion of weak flexibility, where $R =
V$. We further develop this direction by introducing a tool to handle weak
flexibility. We demonstrate this new tool by showing that for every positive
integer $b$ there exists $\epsilon(b)>0$ so that the class of planar graphs
without $K_4, C_5 , C_6 , C_7, B_b$ is weakly $\epsilon(b)$-flexible for lists
of size $4$ (here $K_n$, $C_n$ and $B_n$ are the complete graph, a cycle, and a
book on $n$ vertices, respectively). We also show that the class of planar
graphs without $K_4, C_5 , C_6 , C_7, B_5$ is $\epsilon$-flexible for lists of
size $4$. The results are tight as these graph classes are not even
3-colorable.
