On Minkowski space and finite geometry

Abstract

The main aim of this interdisciplinary paper is to characterize all maps on finite Minkowski space of arbitrary dimension $n$ that map pairs of distinct light-like events into pairs of distinct light-like events. Neither bijectivity of maps nor preservation of light-likeness in the opposite direction, i.e. from codomain to domain, is assumed. We succeed in in many cases, which include the one with $n$ divisible by 4 and the one with $n$ odd and $\geq 9$, by showing that both bijectivity of maps and the preservation of light-likeness in the opposite direction is obtained automatically. In general, the problem of whether there exist non-bijective mappings that map pairs od distinct light-like events into pairs of distinct light-like events is shown to be related to one of the central problems in finite geometry, namely to existence of ovoids in orthogonal polar space. This problem is still unsolved in general despite a huge amount of research done in this area in the last few decades. The proofs are based on the study of a core of an affine polar graph, which yields results that are closely related to the ones obtained previously by Cameron and Kazanidis for the point graph of a polar space

The core of a vertex transitive complementary prism of a lexicographic product

“…Given a graph it is often difficult to decide if it is a core or not. In the case of some graphs with high degree of symmetry and nice combinatorial properties, the decision is equivalent to some of the longstanding open problems in finite geometry [4,25]. A well known core is the Petersen graph, which has many generalizations.…”

The core of a vertex-transitive complementary prism

“…Despite that each graph has its core, which is unique up to isomorphism, it can be often very difficult to determine if a given graph is a core or not (cf. [6,16,31]). From this point of view, graphs that have either high degree of symmetry (i.e.…”

“…Given a corecomplete graph, it can be extremely complicated to decide if the graph is a core or its core is complete. For some graphs, this task is equivalent to some of the longstanding open problems in finite geometry (see [6,31]). A well known core is the Petersen graph, which has both 'large' automorphism group and 'nice' combinatorial properties.…”

A family of non-Cayley cores based on vertex-transitive or strongly regular self-complementary graphs

“…There are other examples of graphs with the same property, which show that the union of the graph families in [12,43,106] is not optimal (cf. [94,95,96,97]). In these cases it is often very difficult to decide, which of the two possibilities regarding the core is true.…”

“…In these cases it is often very difficult to decide, which of the two possibilities regarding the core is true. In fact, for certain classes of graphs this question translates to some of longstanding open problems in finite geometry [12,97]. For some other results related to cores see [40,41,75,99,105].…”