11,213 research outputs found
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
Density of monochromatic infinite paths
For any subset , we define its upper density to be
. We prove
that every -edge-colouring of the complete graph on contains a
monochromatic infinite path, whose vertex set has upper density at least . This improves on results of Erd\H{o}s and
Galvin, and of DeBiasio and McKenney.Comment: Accepted for publication in The Electronic Journal of Combinatoric
Upper density of monochromatic infinite paths
We prove that in every 2-colouring of the edges of KN there exists a monochromatic infinite path P such that V(P) has upper density at least (12+ √ 8)/17 ≈ 0.87226 and further show that this is best possible. This settles a problem of Erdos and Galvi
Partitioning infinite hypergraphs into few monochromatic Berge-paths
Extending a result of Rado to hypergraphs, we prove that for all s, k, t∈ N with k≥ t≥ 2 , the vertices of every r= s(k- t+ 1) -edge-coloured countably infinite complete k-graph can be partitioned into the cores of at most s monochromatic t-tight Berge-paths of different colours. We further describe a construction showing that this result is best possible
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