299 research outputs found
Dominating sets in projective planes
We describe small dominating sets of the incidence graphs of finite
projective planes by establishing a stability result which shows that
dominating sets are strongly related to blocking and covering sets. Our main
result states that if a dominating set in a projective plane of order is
smaller than (i.e., twice the size of a Baer subplane), then
it contains either all but possibly one points of a line or all but possibly
one lines through a point. Furthermore, we completely characterize dominating
sets of size at most . In Desarguesian planes, we could rely on
strong stability results on blocking sets to show that if a dominating set is
sufficiently smaller than 3q, then it consists of the union of a blocking set
and a covering set apart from a few points and lines.Comment: 19 page
The tangent splash in \PG(6,q)
Let B be a subplane of PG(2,q^3) of order q that is tangent to .
Then the tangent splash of B is defined to be the set of q^2+1 points of
that lie on a line of B. In the Bruck-Bose representation of
PG(2,q^3) in PG(6,q), we investigate the interaction between the ruled surface
corresponding to B and the planes corresponding to the tangent splash of B. We
then give a geometric construction of the unique order--subplane determined
by a given tangent splash and a fixed order--subline.Comment: arXiv admin note: substantial text overlap with arXiv:1303.550
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
The Class of Non-Desarguesian Projective Planes is Borel Complete
For every infinite graph we construct a non-Desarguesian projective
plane of the same size as such that and iff . Furthermore, restricted to structures with domain ,
the map is Borel. On one side, this shows that
the class of countable non-Desarguesian projective planes is Borel complete,
and thus not admitting a Ulm type system of invariants. On the other side, we
rediscover the main result of [15] on the realizability of every group as the
group of collineations of some projective plane. Finally, we use classical
results of projective geometry to prove that the class of countable Pappian
projective planes is Borel complete
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