Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G has girth at least six and all lists have size at least three, then there exists an L-coloring respecting at least a constant fraction of the preferences. * Work on this paper was supported by project 17-04611S (Ramsey-like aspects of graph coloring) of Czech Science Foundation.

Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G is triangle‐free and all lists have size at least four, then there exists an L‐coloring respecting at least a constant fraction of the preferences.

Steinberg and Tovey proved that every n‐vertex planar triangle‐free graph has an independent set of size at least ( n + 1 ) ∕ 3, and described an infinite class of tight examples. We show that all n‐vertex planar triangle‐free graphs except for this one infinite class have independent sets of size at least ( n + 2 ) ∕ 3.