177 research outputs found
On Ordinal Invariants in Well Quasi Orders and Finite Antichain Orders
We investigate the ordinal invariants height, length, and width of well quasi
orders (WQO), with particular emphasis on width, an invariant of interest for
the larger class of orders with finite antichain condition (FAC). We show that
the width in the class of FAC orders is completely determined by the width in
the class of WQOs, in the sense that if we know how to calculate the width of
any WQO then we have a procedure to calculate the width of any given FAC order.
We show how the width of WQO orders obtained via some classical constructions
can sometimes be computed in a compositional way. In particular, this allows
proving that every ordinal can be obtained as the width of some WQO poset. One
of the difficult questions is to give a complete formula for the width of
Cartesian products of WQOs. Even the width of the product of two ordinals is
only known through a complex recursive formula. Although we have not given a
complete answer to this question we have advanced the state of knowledge by
considering some more complex special cases and in particular by calculating
the width of certain products containing three factors. In the course of
writing the paper we have discovered that some of the relevant literature was
written on cross-purposes and some of the notions re-discovered several times.
Therefore we also use the occasion to give a unified presentation of the known
results
Rowmotion and generalized toggle groups
We generalize the notion of the toggle group, as defined in [P. Cameron-D.
Fon-der-Flaass '95] and further explored in [J. Striker-N. Williams '12], from
the set of order ideals of a poset to any family of subsets of a finite set. We
prove structure theorems for certain finite generalized toggle groups, similar
to the theorem of Cameron and Fon-der-Flaass in the case of order ideals. We
apply these theorems and find other results on generalized toggle groups in the
following settings: chains, antichains, and interval-closed sets of a poset;
independent sets, vertex covers, acyclic subgraphs, and spanning subgraphs of a
graph; matroids and convex geometries. We generalize rowmotion, an action
studied on order ideals in [P. Cameron-D. Fon-der-Flaass '95] and [J.
Striker-N. Williams '12], to a map we call cover-closure on closed sets of a
closure operator. We show that cover-closure is bijective if and only if the
set of closed sets is isomorphic to the set of order ideals of a poset, which
implies rowmotion is the only bijective cover-closure map.Comment: 26 pages, 5 figures, final journal versio
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
Dense ideals and cardinal arithmetic
From large cardinals we show the consistency of normal, fine,
-complete -dense ideals on for
successor . We explore the interplay between dense ideals, cardinal
arithmetic, and squares, answering some open questions of Foreman
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