180 research outputs found
On the order of countable graphs
A set of graphs is said to be independent if there is no homomorphism between
distinct graphs from the set. We consider the existence problems related to the
independent sets of countable graphs. While the maximal size of an independent
set of countable graphs is 2^omega the On Line problem of extending an
independent set to a larger independent set is much harder. We prove here that
singletons can be extended (``partnership theorem''). While this is the best
possible in general, we give structural conditions which guarantee independent
extensions of larger independent sets. This is related to universal graphs,
rigid graphs and to the density problem for countable graphs
Metric spaces are Ramsey
AbstractWe prove that the class of all ordered finite metric spaces is a Ramsey class. This solves a problem of Kechris, Pestov and Todorćevic
Contractors for flows
We answer a question raised by Lov\'asz and B. Szegedy [Contractors and
connectors in graph algebras, J. Graph Theory 60:1 (2009)] asking for a
contractor for the graph parameter counting the number of B-flows of a graph,
where B is a subset of a finite Abelian group closed under inverses. We prove
our main result using the duality between flows and tensions and finite Fourier
analysis. We exhibit several examples of contractors for B-flows, which are of
interest in relation to the family of B-flow conjectures formulated by Tutte,
Fulkerson, Jaeger, and others.Comment: 22 pages, 1 figur
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