1,149 research outputs found
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.
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
Sizing the length of complex networks
Among all characteristics exhibited by natural and man-made networks the
small-world phenomenon is surely the most relevant and popular. But despite its
significance, a reliable and comparable quantification of the question `how
small is a small-world network and how does it compare to others' has remained
a difficult challenge to answer. Here we establish a new synoptic
representation that allows for a complete and accurate interpretation of the
pathlength (and efficiency) of complex networks. We frame every network
individually, based on how its length deviates from the shortest and the
longest values it could possibly take. For that, we first had to uncover the
upper and the lower limits for the pathlength and efficiency, which indeed
depend on the specific number of nodes and links. These limits are given by
families of singular configurations that we name as ultra-short and ultra-long
networks. The representation here introduced frees network comparison from the
need to rely on the choice of reference graph models (e.g., random graphs and
ring lattices), a common practice that is prone to yield biased interpretations
as we show. Application to empirical examples of three categories (neural,
social and transportation) evidences that, while most real networks display a
pathlength comparable to that of random graphs, when contrasted against the
absolute boundaries, only the cortical connectomes prove to be ultra-short
Hitting minors, subdivisions, and immersions in tournaments
The Erd\H{o}s-P\'osa property relates parameters of covering and packing of
combinatorial structures and has been mostly studied in the setting of
undirected graphs. In this note, we use results of Chudnovsky, Fradkin, Kim,
and Seymour to show that, for every directed graph (resp.
strongly-connected directed graph ), the class of directed graphs that
contain as a strong minor (resp. butterfly minor, topological minor) has
the vertex-Erd\H{o}s-P\'osa property in the class of tournaments. We also prove
that if is a strongly-connected directed graph, the class of directed
graphs containing as an immersion has the edge-Erd\H{o}s-P\'osa property in
the class of tournaments.Comment: Accepted to Discrete Mathematics & Theoretical Computer Science.
Difference with the previous version: use of the DMTCS article class. For a
version with hyperlinks see the previous versio
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