  We say that a graph $H$ is planar unavoidable if there is a planar graph $G$
such that any red/blue coloring of the edges of $G$ contains a monochromatic
copy of $H$, otherwise we say that $H$ is planar avoidable. I.e., $H$ is planar
unavoidable if there is a Ramsey graph for $H$ that is planar. It follows from
the Four-Color Theorem and a result of Gon\c{c}alves that if a graph is planar
unavoidable then it is bipartite and outerplanar. We prove that the cycle on
$4$ vertices and any path are planar unavoidable. In addition, we prove that
all trees of radius at most $2$ are planar unavoidable and there are trees of
radius $3$ that are planar avoidable. We also address the planar unavoidable
notion in more than two colors.

Combinatorics;Discrete Mathematics