On sets of \(n\) points in general position that determine lines that can be pierced by \(n\) points (Q2207602)

A conjecture by \textit{L. Milićevic} [``Classification theorem for strong triangle blocking arrangements'', Preprint, \url{arXiv:1809.08639}] states that if \(P\) and \(R\) are sets of \(n\) points in the plane, the points of \(P\) are in general position, \(P\) and \(R\) are disjoint, and every line connecting two points \(x\), \(y \in P\) contains a point \(r \in R\), then \(P \cup R\) is contained in a cubic curve. The authors prove this conjecture under the additional assumption that \(r\) lies outside the line segment determined by \(x\) and \(y\). As a consequence, if the point set \(P\) contains at least four points and determines precisely \(n\) directions, and \(R\) is taken as line at infinity, it follows that \(P\) is contained in a conic. An open problem posed by \textit{R. Karasev} in [Period. Math. Hung. 78. No. 2, 157--165 (2019; Zbl 1438.14006)] can be rephrased as follows: If three pairwise disjoint sets \(B\), \(G\) and \(R\) of \(n\) points in general position are such that the connecting line of any points from two sets contains a point of the third set, then the union \(B \cup G \cup R\) is contained in a cubic curve. The authors prove this conjecture under the additional assumption that whenever \(b \in B\) and \(g \in G\) there exists a point \(r \in R\) on the connecting line of \(b\) and \(g\) but \emph{outside} the line segment determined by these points. The proofs provide some information on the structure of the point sets: \(P\) as well as \(B \cup G\) must be in convex positions with points of \(B\) and \(G\) alternating on the boundary. Existence of a cubic curve is a consequence of Chasles' Theorem on the nine intersection points of two triples of straight lines.-