28,078 research outputs found
Monadic second-order definable graph orderings
We study the question of whether, for a given class of finite graphs, one can
define, for each graph of the class, a linear ordering in monadic second-order
logic, possibly with the help of monadic parameters. We consider two variants
of monadic second-order logic: one where we can only quantify over sets of
vertices and one where we can also quantify over sets of edges. For several
special cases, we present combinatorial characterisations of when such a linear
ordering is definable. In some cases, for instance for graph classes that omit
a fixed graph as a minor, the presented conditions are necessary and
sufficient; in other cases, they are only necessary. Other graph classes we
consider include complete bipartite graphs, split graphs, chordal graphs, and
cographs. We prove that orderability is decidable for the so called
HR-equational classes of graphs, which are described by equation systems and
generalize the context-free languages
Graph limits and exchangeable random graphs
We develop a clear connection between deFinetti's theorem for exchangeable
arrays (work of Aldous--Hoover--Kallenberg) and the emerging area of graph
limits (work of Lovasz and many coauthors). Along the way, we translate the
graph theory into more classical probability.Comment: 26 page
Countable locally 2-arc-transitive bipartite graphs
We present an order-theoretic approach to the study of countably infinite
locally 2-arc-transitive bipartite graphs. Our approach is motivated by
techniques developed by Warren and others during the study of cycle-free
partial orders. We give several new families of previously unknown countably
infinite locally-2-arc-transitive graphs, each family containing continuum many
members. These examples are obtained by gluing together copies of incidence
graphs of semilinear spaces, satisfying a certain symmetry property, in a
tree-like way. In one case we show how the classification problem for that
family relates to the problem of determining a certain family of highly
arc-transitive digraphs. Numerous illustrative examples are given.Comment: 29 page
Every countable model of set theory embeds into its own constructible universe
The main theorem of this article is that every countable model of set theory
M, including every well-founded model, is isomorphic to a submodel of its own
constructible universe. In other words, there is an embedding that
is elementary for quantifier-free assertions. The proof uses universal digraph
combinatorics, including an acyclic version of the countable random digraph,
which I call the countable random Q-graded digraph, and higher analogues
arising as uncountable Fraisse limits, leading to the hypnagogic digraph, a
set-homogeneous, class-universal, surreal-numbers-graded acyclic class digraph,
closely connected with the surreal numbers. The proof shows that contains
a submodel that is a universal acyclic digraph of rank . The method of
proof also establishes that the countable models of set theory are linearly
pre-ordered by embeddability: for any two countable models of set theory, one
of them is isomorphic to a submodel of the other. Indeed, they are
pre-well-ordered by embedability in order-type exactly .
Specifically, the countable well-founded models are ordered by embeddability in
accordance with the heights of their ordinals; every shorter model embeds into
every taller model; every model of set theory is universal for all
countable well-founded binary relations of rank at most ; and every
ill-founded model of set theory is universal for all countable acyclic binary
relations. Finally, strengthening a classical theorem of Ressayre, the same
proof method shows that if is any nonstandard model of PA, then every
countable model of set theory---in particular, every model of ZFC---is
isomorphic to a submodel of the hereditarily finite sets of . Indeed,
is universal for all countable acyclic binary relations.Comment: 25 pages, 2 figures. Questions and commentary can be made at
http://jdh.hamkins.org/every-model-embeds-into-own-constructible-universe.
(v2 adds a reference and makes minor corrections) (v3 includes further
changes, and removes the previous theorem 15, which was incorrect.
Invariant measures concentrated on countable structures
Let L be a countable language. We say that a countable infinite L-structure M
admits an invariant measure when there is a probability measure on the space of
L-structures with the same underlying set as M that is invariant under
permutations of that set, and that assigns measure one to the isomorphism class
of M. We show that M admits an invariant measure if and only if it has trivial
definable closure, i.e., the pointwise stabilizer in Aut(M) of an arbitrary
finite tuple of M fixes no additional points. When M is a Fraisse limit in a
relational language, this amounts to requiring that the age of M have strong
amalgamation. Our results give rise to new instances of structures that admit
invariant measures and structures that do not.Comment: 46 pages, 2 figures. Small changes following referee suggestion
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