In this talk, I will present the partial order of morphism-extension classes for graphs and connected graphs, and a surprising result for countably infinite connected HH-homogeneous graphs: if they embed an infinite independent set, then the Rado graph is a spanning subgraph. As a corollary, almost all MB-homogeneous graphs are bimorphism-equivalent to the Rado graph.

Joint work with Thomas Coleman and David Hartman.

Random graph processes giving unique limits are, however, rare. Recent joint work with Jeannette Janssen proved the existence of a family of random geometric graphs with unique limits. These graphs arise in the normed space $\ell _{\infty }^{n}$, which consists of $\mathbb{R}^{n}$ equipped with the $L_{\infty }$-norm. Balister, Bollobas, Gunderson, Leader, and Walters used tools from functional analysis to show that these unique limit graphs are deeply tied to the $L_{\infty }$-norm. Precisely, a random geometric graph on any normed, finite-dimensional space not isometric $\ell _{\infty}^{n}$ gives non-isomorphic limits with probability $1$.

With Janssen and Anthony Quas, we have discovered unique limits in infinite dimensional settings including sequences spaces and spaces of continuous functions. We survey these newly discovered infinite random geometric graphs and their properties.

We extend to uncountable languages the studies of the model theory of saturated free algebras in countable languages by [Baldwin, Shelah (1983)], and generalizations by [Pillay, Sklinos (2015)].

A theory \( T \) of cardinality \(\tau\) is \( (\mu,\kappa) \)-almost indiscernible (\(1\le\mu\le\tau<\kappa \)) if \( T \) has a saturated model of power \( \kappa \) which is the algebraic closure of an indiscernible set of \( \mu \)-tuples. \( T \) is \( (\mu,\kappa) \)-almost indiscernible iff \( T \) is \( (\mu,\tau^{+}) \)-almost indiscernible.

Any such theory \( T \) is superstable, stable in all cardinalities \( \lambda\ge\tau \), and non-multidimensional. There is a pairwise orthogonal family \( \mathcal{R} \) of regular types over \( M_{\tau} \), the \( \tau \)-saturated model of cardinality \( \tau \), and a corresponding family \( \mathcal{W} \) of types of maximal weight one sets over \( M_{\tau} \), such that any \( N\succeq M_{\tau} \) is prime over \( M_{\tau} \) and an independent family of realizations of types in \( \mathcal{R} \), and is the algebraic closure of \( M_{\tau} \) and an independent family of realizations of types in \( \mathcal{W} \).

In a saturated free algebra in an uncountable language, the type of a basis element is well-defined. We conjecture that for an arbitrary such theory: it is totally transcendental, one-based, with quantifer elimination down to primitive formulas, and with finite Morley rank.

We have sharper results for theories of modules.

Let $G$ be a finite group, and consider the following decision problem: given a quantified system of equations such as $$\exists x \forall y \exists z \, x^2y^3=z^4xz \wedge y^2z^5 = x \wedge \cdots,$$ is it valid ? Recent developments in the algebraic theory of the computational complexity of quantified constraint satisfaction problems (QCSP) lead us to study the nature of the surjective operations compatible with the group operation, i.e. the group epimomorphisms of the form $f: {G}^n \rightarrow G$. We show, amongst other results, that if $G$ is a quasi-simple group, then all these operations are essentially unary, whence the associated QCSP is PSPACE-complete.\

(Work in progress with Barnaby Martin, University of Durham, UK)

In the 1990s, Archdeacon and Grable and Rodl and Thomas proved that the genus of random graphs behaves very much like the genus of complete graphs.

The talk will outline recent results (joint work with Yifan Jing) about genus embeddings of dense graphs building on the above-mentioned work. These results, which were originally motivated by algorithmic questions, use contemporary notions of quasi-randomness and graph limits, and lead to interesting new problems in topological graph theory.

We compute the ordinal length of the set $\mathbf {I}_{<\omega}(P)$ of finitely generated ideals of a wqo $P$ that is embeddable into $[\omega]^{<\omega}$, the poset of finite subsets of $\omega$. Building on this result, we compute the ordinal length of the set of monomial ideals in $n$ variables. This answers a question of Aschenbrenner and Pong, 2004.

This is a joint work with C.Delhomm\'e (Universit\'e de la R\'eunion) and M.Sobrani (University of Fes, Morocco).

(1) A structure is indivisible if for every partition of it into two parts there exists an embedding of the structure into one of the parts.

The age of a structure is the class of its finite induced substructures up to isomorphism.

(2) A structure is weakly indivisible if for every partition into two parts, for which the age of one of the parts is not equal to the age of the structure, is embeddable into the other part.

(3) A structure is age indivisible if for every partition into two parts the age of one of the parts is equal to the age of the structure.

It turns out that we may describe these polynomials uniformly in terms of $p$. This description allows us to generalize to any integer value of $p$. Taking $p=1$ we recover a classical family of orthogonal polynomials the Morgan-Voyce polynomials which originally arose in a study of electrical resistance in 1959. The Morgan-Voyce polynomials are known to have many connections with the Fibonacci sequence. Our approach yields a new description of the Morgan-Voyce polynomials in terms of an infinite sequence of binary vectors.

Using certain initial segments of this sequence as the vertices of a polytope, we recover the zigzag order polytopes. These polytopes were considered by Stanley and shown to have strong connections with certain elements of the permutation group and with zigzag (or fence) posets. If time permits we will also illustrate a number of other surprising properties of this sequence of binary vectors.

This is joint work with H.E.A. Campbell (UNB).