389 research outputs found
Splitting finite antichains in the homomorphism order
A structural condition is given for finite maximal antichains in the homomorphism order of relational structures to have the splitting property. It turns out that non-splitting antichains appear only at the bottom of the order. Moreover, we examine looseness and finite antichain extension property for some subclasses of the homomorphism poset. Finally, we take a look at cut-points in this order
Dualities and dual pairs in Heyting algebras
We extract the abstract core of finite homomorphism dualities using the
techniques of Heyting algebras and (combinatorial) categories.Comment: 17 pages; v2: minor correction
The Ramsey Theory of Henson graphs
Analogues of Ramsey's Theorem for infinite structures such as the rationals
or the Rado graph have been known for some time. In this context, one looks for
optimal bounds, called degrees, for the number of colors in an isomorphic
substructure rather than one color, as that is often impossible. Such theorems
for Henson graphs however remained elusive, due to lack of techniques for
handling forbidden cliques. Building on the author's recent result for the
triangle-free Henson graph, we prove that for each , the
-clique-free Henson graph has finite big Ramsey degrees, the appropriate
analogue of Ramsey's Theorem.
We develop a method for coding copies of Henson graphs into a new class of
trees, called strong coding trees, and prove Ramsey theorems for these trees
which are applied to deduce finite big Ramsey degrees. The approach here
provides a general methodology opening further study of big Ramsey degrees for
ultrahomogeneous structures. The results have bearing on topological dynamics
via work of Kechris, Pestov, and Todorcevic and of Zucker.Comment: 75 pages. Substantial revisions in the presentation. Submitte
A Special Class of Almost Disjoint Families
The collection of branches (maximal linearly ordered sets of nodes) of the
tree (ordered by inclusion) forms an almost disjoint
family (of sets of nodes). This family is not maximal -- for example, any level
of the tree is almost disjoint from all of the branches. How many sets must be
added to the family of branches to make it maximal? This question leads to a
series of definitions and results: a set of nodes is {\it off-branch} if it is
almost disjoint from every branch in the tree; an {\it off-branch family} is an
almost disjoint family of off-branch sets; {\frak o}=\min\{|{\Cal O}|: {\Cal
O} is a maximal off-branch family. Results concerning include:
(in ZFC) , and (consistent with ZFC) is not
equal to any of the standard small cardinal invariants , ,
, or . Most of these consistency results use
standard forcing notions -- for example, comes from starting with a model of and
adding -many Cohen reals. Many interesting open questions remain,
though -- for example,
Saccharinity
We present a method to iterate finitely splitting lim-sup tree forcings along
non-wellfounded linear orders. We apply this method to construct a forcing
(without using an inaccessible or amalgamation) that makes all definable sets
of reals measurable with respect to a certain (non-ccc) ideal
Generic Large Cardinals and Systems of Filters
We introduce the notion of -system of filters, generalizing the
standard definitions of both extenders and towers of normal ideals. This
provides a framework to develop the theory of extenders and towers in a more
general and concise way. In this framework we investigate the topic of
definability of generic large cardinals properties.Comment: 36 page
Chains, Antichains, and Complements in Infinite Partition Lattices
We consider the partition lattice on any set of transfinite
cardinality and properties of whose analogues do not hold
for finite cardinalities. Assuming the Axiom of Choice we prove: (I) the
cardinality of any maximal well-ordered chain is always exactly ; (II)
there are maximal chains in of cardinality ; (III) if,
for every cardinal , we have , there
exists a maximal chain of cardinality (but ) in
; (IV) every non-trivial maximal antichain in has
cardinality between and , and these bounds are realized.
Moreover we can construct maximal antichains of cardinality for any ; (V) all cardinals of the form
with occur as the number of
complements to some partition , and only these
cardinalities appear. Moreover, we give a direct formula for the number of
complements to a given partition; (VI) Under the Generalized Continuum
Hypothesis, the cardinalities of maximal chains, maximal antichains, and
numbers of complements are fully determined, and we provide a complete
characterization.Comment: 24 pages, 2 figures. Submitted to Algebra Universalis on 27/11/201
- …