A Helly type theorem for abstract projective geometries (Q629834)

A flat in an abstract (axiomatic) projective geometry is a point set \(f\) such that any line \(\ell\) that shares two points with \(f\) is already contained in \(f\). The rank of \(f\) is the minimal cardinality of a point set \(A\) such that \(f\) equals the intersection of all flats containing \(A\). This rank is also defined for the projective geometry itself. The authors prove the following result for projective geometries of rank \(n\): ``A family of linear partitions has non-empty intersection if every \(\lfloor \frac{3n}{2} \rfloor\) members of the family have non-empty intersection.'' Here, a linear partition is the union of a family of flats whose span (the intersection of all flats containing all elements of their union) is of maximal rank. This extends a result by \textit{J. L. Arocha, J. Bracho} and \textit{L. Montejano} [Adv. Math. 213, No.~2, 902--918 (2007; Zbl 1121.52016)] where the same statement was shown to hold true for projective geometries over a (commutative) field.