Combinatorial properties of planar spaces and embeddability
Let S be a planar space i.e. a geometric lattice of rank 4. \textit{J. Kahn} [Math. Z, 175, 219-247 (1980; Zbl 0443.06013)] proved that if all points- residues of S are projective planes and if S satisfies a so called ''bundle axiom'', then S is embeddable in a 3-dimensional projective space. The present paper gives combinatorial conditions implying the embeddability of a planar space as a subset \({\mathcal P}\) of points of some (not necessarily 3-dimensional) projective space, together with the projective lines and planes spanned by points of \({\mathcal P}.\) Suppose each plane V of S is finite and contains \(N(V)+1\) lines through each point, where N(V) depends only of the choice of V. Suppose further that, for every plane V, all lines of V contain at least \((2N(V)+5)/3\) points. Then, the author proofs that N(V) does not depend on V and that S is locally projective and embeddable in a projective space. The proof uses other rather technical conditions implying embeddabiliy for a planar space. Let us mention that Buekenhout's characterization of affine spaces of order \(k\geq 4\) as linear spaces whose planes are affine is a corollary of the main result of the present paper whenever \(k\geq 5\).
embeddability of planar spaces in projective spaces
scientific article; zbMATH DE number 3977824