David Bradley-Williams

Czech Academy of Sciences, Institute of Mathematics | MÚ AV ČR

If G is a graph, A and B its induced subgraphs, and an isomorphism, we say that f is a partial automorphism of G. In 1992, Hrushovski proved that graphs have the extension property for partial automorphisms (EPPA, also called the Hrushovski property), that is, for every finite graph G there is a finite graph H, an EPPA-witness for G, such that G is...

If G is a graph, A, B its induced subgraphs and f : A → B an iso-morphism, we say that f is a partial automorphism of G. In 1992, Hrushovski proved that graphs have the extension property for partial automorphisms (EPPA, also called the Hrushovski property), that is, for every finite graph G there is a finite graph H, its EPPA-witness, such that G...

We prove that Hensel minimal expansions of finitely ramified Henselian valued fields admit spherically complete immediate elementary extensions. More precisely, the version of Hensel minimality we use is 0‐h mix ‐minimality (which, in equi‐characteristic 0, amounts to 0‐h‐minimality).

We give a flexible method for constructing a wide variety of limits of betweenness relations. This unifies work of Adeleke, who constructed a Jordan group preserving a limit of betweenness relations, and Bhattacharjee and Macpherson who gave an alternative method using a Fraïssé-type construction. A key ingredient in their work is the notion of a t...

We introduce a new notion of stratification (``riso-stratifications''), which is entirely canonical and which exists in a variety of settings, including different topological fields like $\mathbb{C}$, $\mathbb{R}$ and $\mathbb{Q}_p$, and also including different o-minimal structures on $\mathbb{R}$. Riso-stratifications are defined directly in term...

We prove that Hensel minimal expansions of finitely ramified Henselian valued fields admit spherically complete elementary extensions.

We give a flexible method for constructing a wide variety of limits of betweenness relations. This unifies work of Adeleke, who constructed a Jordan group preserving a limit of betweenness relations, and Bhattacharjee and Macpherson who gave an alternative method using a Fra\"ss\'e-type construction. A key ingredient in their work is the notion of...

We discuss the Ramsey property, the existence of a stationary independence relation and the coherent extension property for partial isometries (coherent EPPA) for all classes of metrically homogeneous graphs from Cherlin's catalogue, which is conjectured to include all such structures. We show that, with the exception of tree-like graphs, all metri...

We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and we describe their Ramsey expansions.

A partial order is called semilinear iff the upper bounds of each element are linearly ordered and any two elements have a common upper bound. There exists, up to isomorphism, a unique countable semilinear order which is dense, unbounded, binary branching, and without joins. We study the reducts of this semilinear order, that is, the relational str...

We study the notion of weak one-basedness introduced in recent work of Berenstein and Vassiliev. Our main results are that this notion characterizes linearity in the setting of geometric þ-rank 1structures and that lovely pairs of weakly one-based geometric þ-rank 1 structures are weakly one-based with respect to þ-independence. We also study geome...