Nešetřil–Ossona de Mendez conjecture on bounded tree-depth components in 3-colorings of planar graphs

A monochromatic component is a connected component induced by vertices receiving the same color in a vertex coloring. The tree-depth of a graph is the minimum height of a rooted forest whose closure contains the graph.

Nešetřil–Ossona de Mendez conjecture. There exists a constant $t$ such that every planar graph has a vertex coloring with 3 colors in which every monochromatic component has tree-depth at most $t$.

The conjecture is refuted: the construction of $G_3$ in the paper gives planar graphs for which every 3-coloring has a monochromatic component containing a path of arbitrarily large length, and hence of arbitrarily large tree-depth.