Representing embeddability as set inclusion

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


A few steps are made towards a representation theory of embeddability among uncountable graphs. A class of graphs is defined by forbidding some countable configurations which are related to the graph's end-structure. Using uncountable combinatorics, a representation theorem for embeddability in this class is proved, which asserts the existence of a surjective homomorphism from the relation of embeddability over isomorphism types of regular cardinality λ > (greek small letter script kappa)1 onto set inclusion over all subsets of reals or cardinality λ or less. As corollaries we obtain: (1) the complexity of the class in every regular uncountable λ > (greek small letter script kappa)1 is at least λ++ sup{μ(greek small letter script kappa)0 : μ+ < λ}, (2) a characterisation of graphs in the class for which some invariant of the graph has to be inherited by one of fewer than λ subgraphs whose union covers G. Some continuity properties of the homomorphism in the representation theorem are explored and are used to extend the first corollary to all singular cardinals below the first fixed point of second order. The first corollary shows that, contrary to what Shelah has shown for the class of all graphs, the relations of embeddability in the class under discussion is not independent of negations of the generalised continuum hypothesis.

Original languageEnglish
Pages (from-to)257-270
Number of pages14
JournalJournal of the London Mathematical Society
Issue number2
StatePublished - 1 Jan 1998

ASJC Scopus subject areas

  • Mathematics (all)


Dive into the research topics of 'Representing embeddability as set inclusion'. Together they form a unique fingerprint.

Cite this