Property Testing for Point Sets on the Plane

Abstract

A configuration is a point set on the plane, with no three points collinear. Given three non-collinear points p, q and \(r\in \mathbb {R}^2\), let \(\chi (p,q,r)\in \{-1,1\}\), with \(\chi (p,q,r)=1\) if and only if, when we traverse the circle defined by those points in the counterclockwise direction, we encounter the points in the cyclic order \(p,q,r,p,q,r,\dots \). For simplicity, extend \(\chi \) by setting \(\chi (p,q,r)=0\) if p, q and r are not pairwise distinct. Two configurations A and \(B\subset \mathbb {R}^2\) are said to have the same order type if there is a bijection \(\iota :A\rightarrow B\) such that \(\chi (p,q,r)=\chi (\iota (p),\iota (q),\iota (r))\) for all \((p,q,r)\in A^3\). We say that a configuration C contains a copy of a configuration A if there is \(B\subset C\) with A and B of the same order type. Given a configuration F, let \(\mathrm{Forb}(F)\) be the set of configurations that do not contain a copy of F. The distance between two configurations A and B with \(|A|=|B|=n\) is given by