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Flexible list coloring was introduced by Dvo\v{r}\'{a}k, Norin, and Postle in2019. Suppose $0 \leq \epsilon \leq 1$, $G$ is a graph, $L$ is a listassignment for $G$, and $r$ is a function with non-empty domain $D\subseteqV(G)$ such that $r(v) \in L(v)$ for each $v \in D$ ($r$ is called a request of$L$). The triple $(G,L,r)$ is $\epsilon$-satisfiable if there exists a proper$L$-coloring $f$ of $G$ such that $f(v) = r(v)$ for at least $\epsilon|D|$vertices in $D$. $G$ is $(k, \epsilon)$-flexible if $(G,L',r')$ is$\epsilon$-satisfiable whenever $L'$ is a $k$-assignment for $G$ and $r'$ is arequest of $L'$. We improve earlier results in different ways, often obtaininglarger values of $\epsilon$.  In pursuit of the largest $\epsilon$ for which a graph is$(k,\epsilon)$-flexible, we observe that a graph $G$ is not $(k,\epsilon)$-flexible for any $k$ if and only if $\epsilon > 1/ \rho(G)$, where$\rho(G)$ is the Hall ratio of $G$, and we initiate the study of the listflexibility number of a graph $G$, which is the smallest $k$ such that $G$ is$(k,1/ \rho(G))$-flexible. We study relationships and connections between thelist flexibility number, list chromatic number, list packing number, anddegeneracy of a graph.

Flexible list colorings: Maximizing the number of requests satisfied