78 research outputs found
On Partitions of Two-Dimensional Discrete Boxes
Let and be finite sets and consider a partition of the \emph{discrete
box} into \emph{sub-boxes} of the form where and . We say that such a partition has the
-piercing property for positive integers and if every
\emph{line} of the form intersects at least sub-boxes and
every line of the form intersects at least sub-boxes.
We show that a partition of that has the -piercing
property must consist of at least sub-boxes. This bound is nearly sharp (up
to one additive unit) for every and .
As a corollary we get that the same bound holds for the minimum number of
vertices of a graph whose edges can be colored red and blue such that every
vertex is part of red -clique and a blue -clique.Comment: 10 pages, 2 figure
On the Union of Arithmetic Progressions
We show that for every there is an absolute constant
such that the following is true. The union of any
arithmetic progressions, each of length , with pairwise distinct differences
must consist of at least elements. We
observe, by construction, that one can find arithmetic progressions, each
of length , with pairwise distinct differences such that the cardinality of
their union is . We refer also to the non-symmetric case of
arithmetic progressions, each of length , for various regimes of and
Note on the number of edges in families with linear union-complexity
We give a simple argument showing that the number of edges in the
intersection graph of a family of sets in the plane with a linear
union-complexity is . In particular, we prove for intersection graph of a family of
pseudo-discs, which improves a previous bound.Comment: background and related work is now more complete; presentation
improve
An isoperimetric inequality in the universal cover of the punctured plane
AbstractWe find the largest ϵ (approximately 1.71579) for which any simple closed path α in the universal cover R2∖Z2˜ of R2∖Z2, equipped with the natural lifted metric from the Euclidean two-dimensional plane, satisfies L(α)≥ϵA(α), where L(α) is the length of α and A(α) is the area enclosed by α. This generalizes a result of Schnell and Segura Gomis, and provides an alternative proof for the same isoperimetric inequality in R2∖Z2
The number of distinct distances from a vertex of a convex polygon
Erd\H{o}s conjectured in 1946 that every n-point set P in convex position in
the plane contains a point that determines at least floor(n/2) distinct
distances to the other points of P. The best known lower bound due to
Dumitrescu (2006) is 13n/36 - O(1). In the present note, we slightly improve on
this result to (13/36 + eps)n - O(1) for eps ~= 1/23000. Our main ingredient is
an improved bound on the maximum number of isosceles triangles determined by P.Comment: 11 pages, 4 figure
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