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We consider problems where the input is a set of points in the plane and aninteger $k$, and the task is to find a subset $S$ of the input points of size$k$ such that $S$ satisfies some property. We focus on properties that dependonly on the order type of the points and are monotone under point removals. Weshow that not all such problems are fixed-parameter tractable parameterized by$k$, by exhibiting a property defined by three forbidden patterns for whichfinding a $k$-point subset with the property is $\mathrm{W}[1]$-complete and(assuming the exponential time hypothesis) cannot be solved in time$n^{o(k/\log k)}$. However, we show that problems of this type arefixed-parameter tractable for all properties that include all collinear pointsets, properties that exclude at least one convex polygon, and propertiesdefined by a single forbidden pattern.

The Parameterized Complexity of Finding Point Sets with Hereditary Properties