Definition 5.1

We define a property of configurations to be something that is true of some configurations and false for others, but whose truth or falsity depends only on the configuration and not on the set of points that realizes it. A property is monotone if, whenever a configuration has the property, so do all its subconfigurations.

Example 5.2

Containing the vertices of a convex quadrilateral is not a monotone property, but not containing a convex quadrilateral ismonotone.

Definition 5.3

A parameter of configurations is a function from configurations to nonnegative integers. It is monotone if removing points from a configuration cannot cause the function value to increase.

Example 5.4

The size |S| of a configuration S is amonotone parameter, because all realizations of S have the same number of points, and because taking away points reduces the size.

Example 5.5

Whenever ρ is a monotone parameter of configurations, and k is a positive integer, then the property of a configuration S of having ρ(S) < k is a monotone property of configurations. For instance, the property of having size at most four is monotone.

Definition 5.6

If P is a monotone property of configurations, then the obstacles of P are the configurations that do not have P, but for which all proper subconfigurations do have P. P has bounded obstacles if there are finitely many obstacles for P.