User:Katie Thompson

From Young Set Theory Network

Katie Thompson

Institut für diskrete Mathematik und Geometrie
Technische Universität Wien
Austria
http://www.logic.univie.ac.at/~thompson

My research centres around the classification of relational structures (e.g. orders, graphs) and those with extra topological structure (e.g. ordered spaces, Boolean algebras) via the embeddability relation. An embedding is generally an injective structure-preserving map between structures. For example, for linear orders, the ordering is preserved in embeddings. The embeddability relation, $A \leq B$ iff $A$ can be embedded into $B$ , is a quasi-ordering of the structures. I study two aspects of this relation: universal structures, those on the top of the embeddability quasi-order, and internal properties of the embeddability structure.

Universal models are structures which embed all other structures (of the same theory) of the same size. For first-order definable theories, it is known via model theoretic arguments that universal models exist in all cardinals above the size of the language. Considering theories which are not first-order definable and even first-order theories in the absence of GCH usually requires set-theoretic techniques.

When GCH fails, it is often independent whether structures may have universal models in uncountable cardinals. In fact, the techniques used to prove universality results in these cases include forcing and infinite combinatorics. For instance, it is an open question whether there is a model of not CH in which there is a universal triangle-free graph at $\aleph_1$ . However, I have recently constructed a model in which CH fails and $2^{\aleph_1}$ is large in which there is a small universal family for triangle-free graphs, namely a family of size $\aleph_2$ which embeds all other triangle-free graphs.

When GCH holds, I have considered structures like posets or trees which omit large chains and graphs which omit large cliques. When "large" is some infinite cardinal, then the theory of these structures is not first-order definable. As an example of such a result, I have proved that there is a universal model for trees of size $\lambda$ which omit $\kappa$ -branches if and only if $\lambda$ is a (strong) limit cardinal with $\lambda > \kappa \geq \aleph_0$ and cf $(\kappa) >$ cf $(\lambda)$ . Also, I am studying structures like linearly ordered spaces (whose theory is also not first-order definable) under continuous order-preserving embeddings.

I also consider internal properties that the embedding quasi-order may have. For instance, Laver proved that countable linear orders form a well-quasi-order under the embeddability relation. A well-quasi-order is a quasi-order which has no infinite antichains and no infinite descending sequences. Laver's result is proved using Hausdorff's constructive hierarchy of scattered orders (those that do not embed the rationals). We would like to extend such a result to uncountable linear orders, but the notions of "scattered" and "dense" diverge in this case. Namely for uncountable $\kappa$ , we may consider $\kappa$ -dense linear orders to be those in which between every two points, there are $\kappa$ -many points in between and $\kappa$ -saturated to be those in which between every two sets of size $< \kappa$ , there is a point in between. So far, we have a Hausdorff-like constructive hierarchy only for those linear orders which do not embed a $\kappa$ -dense order.

I am also interested in the antichain structure of embedding quasi-orders. In recent work with Philipp Schlicht, we have constructed an antichain of $\kappa$ -trees of size $\kappa$ for cf $(\kappa) = \omega$ in the embedding quasi-order where the embeddings only preserve strict order (not necessarily injective).

For a list of my papers, see
http://www.logic.univie.ac.at/~thompson/pubs.html

User:Katie Thompson

From Young Set Theory Network

Katie Thompson

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