An "unavoidable set" means the same for multivariate splines as for the 4 color map problem.

It is true but not trivial statement that any triangulation can be built by adding one flap or fill at a time, and maintaining a triangulation at every step.

A corresponding statement is not true for three dimensional
triangulations (i.e., tesselations of a three dimensional
domain by tetrahedra).
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[21-Apr-1997]