Crossing-free segments and triangles in point configurations (Q5954240)

In 3-space a triangle in an \(n\)-configuration (a set of \(n\) points in general position) is free if it is not crossed by any other triangle in the configuration. In any configuration of \(n\geq 6\) points in 3-space, there are at most \(3n-6\) free triangles and this result is the best possible. In a \(d\)-dimensional configuration, an \(i\)-simplex \((i<d)\) is \(j\)-free \((j<d)\) if it does not intersect any \(j\)-simplex of the configuration.NEWLINENEWLINENEWLINEFor configuration \({\mathcal V}\), let \(f_d^{i,j} ({\mathcal V})\) be the maximum number of \(j\)-free \(i\)-simplices in a \(d\)-dimensional \(n\)-configuration. Some bounds of this function are obtained for special cases of this more general setting.