Containment of arbitrary 6-chromatic graphs

Establish whether the visibility graph of every sufficiently large finite planar point set with no four collinear points contains a copy of every fixed graph H with chromatic number six, without assuming that H has a color-critical edge.

Background

The paper proves that every fixed graph with chromatic number at most five occurs in the visibility graph of every sufficiently large planar point set with no four collinear points. It also proves the analogous result for every 6-chromatic graph possessing a color-critical edge.

The remaining issue is whether the color-critical-edge hypothesis is necessary. The paper identifies 6-chromatic graphs such as K_{1,2,2,2,2,2} and K_{2,2,2,2,2,2} as lying outside the scope of the theorem because they have no color-critical edge.