Two-graphs and skew two-graphs in finite geometries
Let \(X\) be a set of cardinality \(v\geq 3\). A two-graph on \(X\) is a set, \(\Delta\), of 3-subsets of \(X\) such that every 4-subset of \(X\) contains an even number (i.e., 0, 2 or 4) of triples from \(\Delta\). A skew (oriented) two-graph on \(X\) is a set, \(\nabla\), of 3-cycles in \(\text{Sym}(X)\), such that precisely one of any 3-cycle or its inverse belongs to \(\nabla\), and for any 4-subset of \(X\), \(\nabla\) contains an even number of the 3-cycles that can be formed from these four elements. Given a 2-subset of \(X\) (resp. an ordered pair of distinct elements of \(X\)) its degree is the number of 3-subsets of \(\Delta\) (resp. 3-cycles of \(\nabla\)) which contain it. The multiset of the number of 2-subsets (resp. ordered pairs) of each degree is called the fingerprint of \(\Delta\) (resp. \(\nabla\)). In this paper, fingerprints are used as isomorphic invariants for translation planes and \(m\)-systems (including ovoids and spreads) of polar spaces in odd characteristic. The adventage of using fingerprints in this way is that they are easily and quickly computed, however, they are not generally complete, i.e., able to distinguish between any two inequivalent examples. The question of completeness of fingerprint invariants for ovoids in \(O_{2r}^+(q)\) (for \(q\) odd) is open for \(r\geq 3\), but is false for \(r= 2\) (Theorem 7.3). The author also examines caps in polar spaces whose associated two-graphs are trivial, classifying these in a small case and obtaining a general bound on the size of the cap. These results are used in obtaining a bound on the size of a family of doubly intersecting circles in classical Laguerre planes of odd order.
scientific article; zbMATH DE number 803759
scientific article