3,029 research outputs found
The weakness of the pigeonhole principle under hyperarithmetical reductions
The infinite pigeonhole principle for 2-partitions ()
asserts the existence, for every set , of an infinite subset of or of
its complement. In this paper, we study the infinite pigeonhole principle from
a computability-theoretic viewpoint. We prove in particular that
admits strong cone avoidance for arithmetical and
hyperarithmetical reductions. We also prove the existence, for every
set, of an infinite low subset of it or its complement. This
answers a question of Wang. For this, we design a new notion of forcing which
generalizes the first and second-jump control of Cholak, Jockusch and Slaman.Comment: 29 page
Minimal forbidden sets for degree sequence characterizations
Given a set F of graphs, a graph G is F-free if G does not contain any member of as an induced subgraph. A set F is degree-sequence-forcing (DSF) if, for each graph G in the class C of -free graphs, every realization of the degree sequence of G is also in C. A DSF set is minimal if no proper subset is also DSF. In this paper, we present new properties of minimal DSF sets, including that every graph is in a minimal DSF set and that there are only finitely many DSF sets of cardinality k. Using these properties and a computer search, we characterize the minimal DSF triples
On 2-switches and isomorphism classes
A 2-switch is an edge addition/deletion operation that changes adjacencies in
the graph while preserving the degree of each vertex. A well known result
states that graphs with the same degree sequence may be changed into each other
via sequences of 2-switches. We show that if a 2-switch changes the isomorphism
class of a graph, then it must take place in one of four configurations. We
also present a sufficient condition for a 2-switch to change the isomorphism
class of a graph. As consequences, we give a new characterization of matrogenic
graphs and determine the largest hereditary graph family whose members are all
the unique realizations (up to isomorphism) of their respective degree
sequences.Comment: 11 pages, 6 figure
On the noncommutative geometry of tilings
This is a chapter in an incoming book on aperiodic order. We review results
about the topology, the dynamics, and the combinatorics of aperiodically
ordered tilings obtained with the tools of noncommutative geometry
Dynamics and the Emergence of Geometry in an Information Mesh
The idea of a graph theoretical approach to modeling the emergence of a
quantized geometry and consequently spacetime, has been proposed previously,
but not well studied. In most approaches the focus has been upon how to
generate a spacetime that possesses properties that would be desirable at the
continuum limit, and the question of how to model matter and its dynamics has
not been directly addressed. Recent advances in network science have yielded
new approaches to the mechanism by which spacetime can emerge as the ground
state of a simple Hamiltonian, based upon a multi-dimensional Ising model with
one dimensionless coupling constant. Extensions to this model have been
proposed that improve the ground state geometry, but they require additional
coupling constants. In this paper we conduct an extensive exploration of the
graph properties of the ground states of these models, and a simplification
requiring only one coupling constant. We demonstrate that the simplification is
effective at producing an acceptable ground state. Moreover we propose a scheme
for the inclusion of matter and dynamics as excitations above the ground state
of the simplified Hamiltonian. Intriguingly, enforcing locality has the
consequence of reproducing the free non-relativistic dynamics of a quantum
particle
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