Graphs of maximum average degree less than 11/3 are flexibly $4$-choosable

We consider the flexible list coloring problem, in which we have a graph G, a color list assignment L:V(G) → 2N, and a set U ⊆ V(G) of vertices such that each u ∈ U has a preferred color p(u) ∈ L(u). Given a constant ε > 0, the problem asks for an L-coloring of G in which at least ε |U| vertices in U receive their preferred color. We use a method of reducible subgraphs to approach this problem. We develop a vertex-partitioning tool that, when used with a new reducible subgraph framework, allows us to define large reducible subgraphs. Using this new tool, we show that if G has maximum average degree less than 11/3, a list $L(v)$ of size $4$ at each v ∈ V(G), and a set U ⊆ V(G) of vertices with preferred colors, then there exists an L-coloring of G for which at least 2⁻¹⁴⁵ |U| vertices of U receive their preferred color.