1,806 research outputs found
Some non-existence results for distance- ovoids in small generalized polygons
We give a computer-based proof for the non-existence of distance- ovoids
in the dual split Cayley hexagon .
Furthermore, we give upper bounds on partial distance- ovoids of
for .Comment: 10 page
Non-intersecting Ryser hypergraphs
A famous conjecture of Ryser states that every -partite hypergraph has
vertex cover number at most times the matching number. In recent years,
hypergraphs meeting this conjectured bound, known as -Ryser hypergraphs,
have been studied extensively. It was recently proved by Haxell, Narins and
Szab\'{o} that all -Ryser hypergraphs with matching number are
essentially obtained by taking disjoint copies of intersecting -Ryser
hypergraphs. Abu-Khazneh showed that such a characterisation is false for by giving a computer generated example of a -Ryser hypergraph with whose vertex set cannot be partitioned into two sets such that we have an
intersecting -Ryser hypergraph on each of these parts. Here we construct new
infinite families of -Ryser hypergraphs, for any given matching number , that do not contain two vertex disjoint intersecting -Ryser
subhypergraphs.Comment: 8 pages, some corrections in the proof of Lemma 3.6, added more
explanation in the appendix, and other minor change
Characterizations of the Suzuki tower near polygons
In recent work, we constructed a new near octagon from certain
involutions of the finite simple group and showed a correspondence
between the Suzuki tower of finite simple groups, , and the tower of near polygons, . Here we characterize
each of these near polygons (except for the first one) as the unique near
polygon of the given order and diameter containing an isometrically embedded
copy of the previous near polygon of the tower. In particular, our
characterization of the Hall-Janko near octagon is similar to an
earlier characterization due to Cohen and Tits who proved that it is the unique
regular near octagon with parameters , but instead of regularity
we assume existence of an isometrically embedded dual split Cayley hexagon,
. We also give a complete classification of near hexagons of
order and use it to prove the uniqueness result for .Comment: 20 pages; some revisions based on referee reports; added more
references; added remarks 1.4 and 1.5; corrected typos; improved the overall
expositio
A new near octagon and the Suzuki tower
We construct and study a new near octagon of order which has its
full automorphism group isomorphic to the group and which
contains copies of the Hall-Janko near octagon as full subgeometries.
Using this near octagon and its substructures we give geometric constructions
of the -graph and the Suzuki graph, both of which are strongly
regular graphs contained in the Suzuki tower. As a subgeometry of this octagon
we have discovered another new near octagon, whose order is .Comment: 24 pages, revised version with added remarks and reference
On semi-finite hexagons of order containing a subhexagon
The research in this paper was motivated by one of the most important open
problems in the theory of generalized polygons, namely the existence problem
for semi-finite thick generalized polygons. We show here that no semi-finite
generalized hexagon of order can have a subhexagon of order .
Such a subhexagon is necessarily isomorphic to the split Cayley generalized
hexagon or its point-line dual . In fact, the employed
techniques allow us to prove a stronger result. We show that every near hexagon
of order which contains a generalized hexagon of
order as an isometrically embedded subgeometry must be finite. Moreover, if
then must also be a generalized hexagon, and
consequently isomorphic to either or the dual twisted triality hexagon
.Comment: 21 pages; new corrected proofs of Lemmas 4.6 and 4.7; earlier proofs
worked for generalized hexagons but not near hexagon
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