This is the second of two volumes by Professor Cherlin presenting the state of the art in the classification of homogeneous structures in binary languages and related problems in the intersection of model theory and combinatorics. Researchers and graduate students in the area will find in these volumes many far-reaching results and interesting new research directions to pursue. This volume continues the analysis of the first volume to 3-multi-graphs and 3-multi-tournaments, expansions of graphs and tournaments by the addition of a further binary relation. The opening chapter provides an overview of the volume, outlining the relevant results and conjectures. The author applies and extends the results of Volume I to obtain a detailed catalogue of such structures and a second classification conjecture. The book ends with an appendix exploring recent advances and open problems in the theory of homogeneous structures and related subjects.

We study Ramsey expansions of certain homogeneous 3-hypertournaments. We show that they exhibit an interesting behaviour and, in one case, they seem not to submit to current gold-standard methods for obtaining Ramsey expansions. This makes these examples very interesting from the point of view of structural Ramsey theory as there is a large demand for novel examples.

We give a bibliography of works relating to homogeneous structures in the sense of Fra\iss\'e, and related topics, mainly through 2016, with some narrow updating through 2021. We first give a list arranged by topics, with references to the main bibliogrphay, which follows.

We classify countable metrically homogeneous graphs of diameter 3.

Let C be a finite connected graph for which there is a countable universal C-free graph, and whose tree of blocks is a path. Then the blocks of C are complete. This generalizes a result of Füredi and Komjáth, and fits naturally into a set of conjectures regarding the existence of countable C-free graphs, with C an arbitrary finite connected graph.

The relational complexity \(\rho (X,G)\) of a finite permutation group is the least k for which the group can be viewed as an automorphism group acting naturally on a homogeneous relational system whose relations are k-ary (an explicit permutation group theoretic version of this definition is also given). In the context of primitive permutation groups, the natural questions are (a) rough estimates, or (preferably) precise values for \(\rho \) in natural cases; and (b) a rough determination of the primitive permutation groups with \(\rho \) either very small (bounded) or very large (much larger than the logarithm of the degree). The rough version of (a) is relevant to (b). Our main result is an explicit characterization of the binary (\(\rho =2\)) primitive affine permutation groups. We also compute the precise relational complexity of \({{\mathrm{Alt}}}_n\) acting on k-sets, correcting (Cherlin in Sporadic homogeneous structures. In: The Gelfand Mathematical Seminars, 1996–1999, pp. 15–48, Birkhäuser 2000, Example 5).

Мы рассматриваем группы, действующие регулярно на графах Хенсона $\Gamma_n$, отвечая на вопрос Питера Камерона, а также изучаем смежные вопросы.

We discuss groups acting regularly on the Henson graphs Γ n , answering a question posed by Peter Cameron, and we explore a number of related questions.

1. Morley, Marsh, Baldwin/Lachlan, Zilber; and Macintyre 2 2. The Borovik Program: CFSG and tameness 3 2.1. The setting: tame or not 3 2.2. The basic framework: even, odd, mixed, degenerate 4 3. Mixed type 5 3.1. Strong and weak embedding 5 3.2. Toward weak embedding: U(G) 6 3.3. The subgroups B(G) and D(G) 6 3.4. The strategy 7 3.5. The curious incident of the dog in the night 8 3.6. Weak embedding 9 3.7. Strong Embedding 11 4. The weak embedding theorem 12 5. Minimal simple groups 13 6. Strongly embedded Borels in the minimal simple nontame case 16 7. Non-nilpotent strongly embedded Borel subgroups 18 7.1. Preliminary remarks and notation 18 7.2. The argument 19 8. Appendix: some definitions 19 References 21

AbstractWe study definable irreducible actions of SL2on an abelian group of Morley rank ≤ 3rkand prove they are rational representations of the group.

We take up Peter Cameron's problem of the classification of count- ably infinite graphs which are homogeneous as metric spaces in the graph metric (Cam98). We give an explicit catalog of the known examples, together with results supporting the conjecture that the catalog may be complete, or nearly so. We begin in Part I with a presentation of Fra¨osse's theory of amalgamation classes and the classification of homogeneous structures, with emphasis on the case of homogeneous metric spaces, from the discovery of the Urysohn space to the connection with topological dynamics developed in (KPT05). We then turn to a discussion of the known metrically homogeneous graphs in Part II. This includes a 5-parameter family of homogeneous metric spaces whose connections with topological dynamics remain to be worked out. In the case of diameter 4, we find a variety of examples buried in the tables at the end of (Che98), which we decode and correlate with our catalog. In the final Part we revisit an old chestnut from the theory of homoge- neous structures, namely the problem of approximating the generic triangle free graph by finite graphs. Little is known about this, but we rephrase the problem more explicitly in terms of finite geometries. In that form it leads to questions that seem appropriate for design theorists, as well as some ques- tions that involve structures small enough to be explored computationally. We also show, following a suggestion of Peter Cameron (1996), that while strongly regular graphs provide some interesting examples, one must look beyond this class in general for the desired approximations.

We discuss two combinatorial problems concerning classes of finite or countable structures of combinatorial type. We consider classes determined by a finite set of finite constraints (forbidden substructures). Questions about such classes of structures are naturally viewed as algorithmic decision problems, taking the finite set of constraints as the input. While the two problems we consider have been studied in a number of natural contexts, it remains far from clear whether they are decidable in their general form. This broad question leads to a number of more concrete problems. We discuss twelve open problems of varying levels of concreteness, and we point to the “Hairy Ball Problem” as a particularly concrete problem, which we give first in direct model theoretic terms, and then decoded as an explicit graph theoretic problem.

The Algebraicity Conjecture treats model-theoretic foundations of algebraic group theory. It states that any simple group of finite Morley rank is an algebraic group over an algebraically closed field. In the mid-1990s a view was consolidated that this project falls into four cases of different flavour: even type, mixed type, odd type, and degenerate type. This book contains a proof of the conjecture in the first two cases, and much more besides: insight into the current state of the other cases (which are very much open), applications for example to permutation groups of finite Morley rank, and open questions. The book will be of interest to both model theorists and group theorists: techniques from the classification of finite simple groups (CFSG), and from other aspects of group theory (e.g., black box groups in computational group theory, and the theory of Tits buildings, especially of generalised polygons) play a major role. The techniques used are primarily group theoretic, but the history and motivation are more model theoretic. The origins of the conjecture, as with much modern model theory, lie in Morley’s Theorem: this states that if T is a complete theory in a countable first order language L (that is, T is a maximally consistent set of L-sentences) and T is κ-categorical for some uncountable cardinal κ, then T is κcategorical for all uncountable κ, that is, T is uncountably categorical. Here, the theory T is κ-categorical if all its models of size κ are isomorphic. Morley’s proof in [8] introduces the notion of Morley rank, an abstract dimension notion for definable sets (i.e., solution sets of formulas). In an instructive example, where T is the theory of an algebraically closed field K, definable sets are exactly the same as constructible sets (by quantifier elimination), and the Morley rank of a constructible set is just the algebraic-geometric dimension of its Zariski closure. Morley rank is ordinal valued: the Morley rank of a definable set X is at least α+1 if it is possible (at least after moving to an elementary extension) to partition X into infinitely many definable sets of rank at least α, and the definition at limit ordinals is the natural one. Morley showed that in an uncountably categorical theory, all definable sets have ordinal-valued Morley rank. In particular, such theories are stable (in fact, ω-stable). The rank was later shown by Baldwin to be finite. A new proof of Morley’s Theorem, yielding additional information, was given by Baldwin and Lachlan [1]. They showed that any uncountably categorical structure is “coordinatised” by a strongly minimal set, that is, a definable set all of whose definable subsets are finite or cofinite, uniformly in parameters. The form of coordinatisation was later refined by Zilber in his “Ladder theorem”; he also formulated conjectures on the structure of strongly minimal sets: obvious examples are pure sets (trivial geometry), vector spaces (locally modular geometry) and algebraically closed fields, and Zilber conjectured that any strongly minimal set with nonlocally modular geometry is very close to an algebraically closed field. This was proved false by Hrushovski [6]. With a new and delicate amalgamation technique

We continue the analysis of metrically homogeneous graphs, taking up where the catalog left off. We give some general results on the structure of Γi, and show that the metrically homogeneous graphs of diameter 3 are all in the catalog.

We prove several results about groups of finite Morley rank without unipotent p-torsion: p-torsion always occurs inside tori, Sylow p-subgroups are conjugate, and p is not the minimal prime divisor of our approximation to the "Weyl group". These results are quickly finding extensive applications within the classification project.

In this article we consider representations of SL2 which are interpretable in nite Morley rank theories, meaning that inside a universe of nite Morley rank we shall study the following de nable objects: a group G isomorphic to SL2, an abelian group V , and an action of G on V ; V is thus a de nable G-module on which G acts de nably. Our goal will be to identify V with a standard Gmodule, under an assumption on its Morley rank. (A word on this notion of rank will be said shortly, after we have stated the results.) It will be convenient to work with a faithful representation, possibly replacing SL2 by the quotient PSL2, and we shall write G ' (P)SL2 to cover both cases.