1,309 research outputs found
On some points-and-lines problems and configurations
We apply an old method for constructing points-and-lines configurations in
the plane to study some recent questions in incidence geometry.Comment: 14 pages, numerous figures of point-and-line configurations; to
appear in the Bezdek-50 special issue of Periodica Mathematica Hungaric
Finite height lamination spaces for surfaces
We describe spaces of essential finite height (measured) laminations in a
surface using a parameter space we call , an ordered semi-ring.
We show that for every finite height essential lamination in , there is
an action of on an -tree dual to the lift of to the
universal cover of
Transversals in a collections of trees
Let be a fixed family of graphs on vertex set and
be a collection of elements in . We investigated the
transversal problem of finding the maximum value of when
contains no rainbow elements in . Specifically, we
determine the exact values when is a family of stars or a family
of trees of the same order with dividing . Further, all the
extremal cases for are characterized.Comment: 16pages,2figure
Red-blue clique partitions and (1-1)-transversals
Motivated by the problem of Gallai on -transversals of -intervals,
it was proved by the authors in 1969 that if the edges of a complete graph
are colored with red and blue (both colors can appear on an edge) so that there
is no monochromatic induced and then the vertices of can be
partitioned into a red and a blue clique. Aharoni, Berger, Chudnovsky and Ziani
recently strengthened this by showing that it is enough to assume that there is
no induced monochromatic and there is no induced in {\em one of the
colors}. Here this is strengthened further, it is enough to assume that there
is no monochromatic induced and there is no on which both color
classes induce a .
We also answer a question of Kaiser and Rabinovich, giving an example of six
-convex sets in the plane such that any three intersect but there is no
-transversal for them
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