4,241 research outputs found
Wide scattered spaces and morasses
We show that it is relatively consistent with ZFC that 2^omega is arbitrarily
large and every sequence s=(s_i:i<omega_2) of infinite cardinals with
s_i<=2^omega is the cardinal sequence of some locally compact scattered space.Comment: 14 page
Decompositions of edge-colored infinite complete graphs into monochromatic paths
An -edge coloring of a graph or hypergraph is a map . Extending results of Rado and answering questions of Rado,
Gy\'arf\'as and S\'ark\"ozy we prove that
(1.) the vertex set of every -edge colored countably infinite complete
-uniform hypergraph can be partitioned into monochromatic tight paths
with distinct colors (a tight path in a -uniform hypergraph is a sequence of
distinct vertices such that every set of consecutive vertices forms an
edge),
(2.) for all natural numbers and there is a natural number such
that the vertex set of every -edge colored countably infinite complete graph
can be partitioned into monochromatic powers of paths apart from a
finite set (a power of a path is a sequence of
distinct vertices such that implies that is an
edge),
(3.) the vertex set of every -edge colored countably infinite complete
graph can be partitioned into monochromatic squares of paths, but not
necessarily into ,
(4.) the vertex set of every -edge colored complete graph on
can be partitioned into monochromatic paths with distinct colors
Essentially disjoint families, conflict free colorings and Shelah's Revised GCH
Using Shelah's revised GCH theorem we prove that if mu<beth_omega <= lambda
are cardinals, then every mu-almost disjoint subfamily B of
[lambda]^{beth_omega} is essentially disjoint, i.e. for each b from B there is
a subset f(b) of b of size < |b| such that the family {b-f(b) b in B} is
disjoint.
We also show that if mu<=kappa<=lambda, and kappa is infinite, and (x) every
mu-almost disjoint subfamily of [lambda]^kappa is essentially disjoint, then
(xx) every mu-almost disjoint family B of subsets of lambda with |b|>=kappa for
all b from B has a conflict-free colorings with kappa colors.
Putting together these results we obtain that if mu<beth_omega<=lambda, then
every mu-almost disjoint family B of subsets of lambda with |b|>=beth_omega for
all b from B has a conflict-free colorings with beth_omega colors.
To yield the above mentioned results we also need to prove a certain
compactness theorem concerning singular cardinals.Comment: 10 pages, minor correction
Elementary submodels in infinite combinatorics
The usage of elementary submodels is a simple but powerful method to prove
theorems, or to simplify proofs in infinite combinatorics. First we introduce
all the necessary concepts of logic, then we prove classical theorems using
elementary submodels. We also present a new proof of Nash-Williams's theorem on
cycle-decomposition of graphs, and finally we improve a decomposition theorem
of Laviolette concerning bond-faithful decompositions of graphs
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