2,185 research outputs found
Regular finite decomposition complexity
We introduce the notion of regular finite decomposition complexity of a metric family. This generalizes Gromov's finite asymptotic dimension and is motivated by the concept of finite decomposition complexity (FDC) due to Guentner, Tessera and Yu. Regular finite decomposition complexity implies FDC and has all the permanence properties that are known for FDC, as well as a new one called Finite Quotient Permanence. We show that for a collection containing all metric families with finite asymptotic dimension all other permanence properties follow from Fibering Permanence
Unsplittable coverings in the plane
A system of sets forms an {\em -fold covering} of a set if every point
of belongs to at least of its members. A -fold covering is called a
{\em covering}. The problem of splitting multiple coverings into several
coverings was motivated by classical density estimates for {\em sphere
packings} as well as by the {\em planar sensor cover problem}. It has been the
prevailing conjecture for 35 years (settled in many special cases) that for
every plane convex body , there exists a constant such that every
-fold covering of the plane with translates of splits into
coverings. In the present paper, it is proved that this conjecture is false for
the unit disk. The proof can be generalized to construct, for every , an
unsplittable -fold covering of the plane with translates of any open convex
body which has a smooth boundary with everywhere {\em positive curvature}.
Somewhat surprisingly, {\em unbounded} open convex sets do not misbehave,
they satisfy the conjecture: every -fold covering of any region of the plane
by translates of such a set splits into two coverings. To establish this
result, we prove a general coloring theorem for hypergraphs of a special type:
{\em shift-chains}. We also show that there is a constant such that, for
any positive integer , every -fold covering of a region with unit disks
splits into two coverings, provided that every point is covered by {\em at
most} sets
Maximum union-free subfamilies
An old problem of Moser asks: how large of a union-free subfamily does every
family of m sets have? A family of sets is called union-free if there are no
three distinct sets in the family such that the union of two of the sets is
equal to the third set. We show that every family of m sets contains a
union-free subfamily of size at least \lfloor \sqrt{4m+1}\rfloor - 1 and that
this bound is tight. This solves Moser's problem and proves a conjecture of
Erd\H{o}s and Shelah from 1972. More generally, a family of sets is
a-union-free if there are no a+1 distinct sets in the family such that one of
them is equal to the union of a others. We determine up to an absolute
multiplicative constant factor the size of the largest guaranteed a-union-free
subfamily of a family of m sets. Our result verifies in a strong form a
conjecture of Barat, F\"{u}redi, Kantor, Kim and Patkos.Comment: 10 page
Generalized strongly increasing semigroups
In this work we present a new class of numerical semigroups called
GSI-semigroups. We see the relations between them and others families of
semigroups and we give explicitly their set of gaps. Moreover, an algorithm to
obtain all the GSI-semigroups up to a given Frobenius number is provided and
the realization of positive integers as Frobenius numbers of GSI-semigroups is
studied
Helly numbers of Algebraic Subsets of
We study -convex sets, which are the geometric objects obtained as the
intersection of the usual convex sets in with a proper subset
. We contribute new results about their -Helly
numbers. We extend prior work for , , and ; we give sharp bounds on the -Helly numbers in
several new cases. We considered the situation for low-dimensional and for
sets that have some algebraic structure, in particular when is an
arbitrary subgroup of or when is the difference between a
lattice and some of its sublattices. By abstracting the ingredients of Lov\'asz
method we obtain colorful versions of many monochromatic Helly-type results,
including several colorful versions of our own results.Comment: 13 pages, 3 figures. This paper is a revised version of what was
originally the first half of arXiv:1504.00076v
- …