arXiv subjects
Publications and source records attributed to Vera Koponen.
Suppose that M is an infinite structure with finite relational vocabulary such that every relation symbol has arity at most 2. If M is simple and homogeneous then its complete theory is supersimple with finite SU-rank which cannot exceed the number of complete 2-types over the empty set.
We study definable sets $D$ of SU-rank 1 in $M^{eq}$, where $M$ is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such $D$ can be seen as a `canonically embedded structure', which inherits all relations on $D$ which are definable in $M^{eq}$, and has no other definable relations. Our results imply that if no relation symbol of the language of $M$ has arity higher than 2, then there is a close relationship between triviality of dependence and $D$ being a reduct of a binary random structure. Somewhat more preciely: (a) if for every $n \geq 2$, every $n$-type $p(x_1, ..., x_n)$ which is realized in $D$ is determined by its sub-2-types $q(x_i, x_j) \subseteq p$, then the algebraic closure restricted to $D$ is trivial; (b) if $M$ has trivial dependence, then $D$ is a reduct of a binary random structure.
We work with a finite relational vocabulary with at least one relation symbol with arity at least 2. Fix any integer $m > 1$. For almost all finite structures (labelled or unlabelled) such that at least $m$ elements are moved by some automorphisms, the automorphism group is $(Z_2)^i$ for some $i \leq (m+1)/2$; and if some relation symbol has arity at least 3, then the automorphism group is almost always $Z_2$.
A systematic study is made, for an arbitrary finite relational language with at least one symbol of arity at least 2, of classes of nonrigid finite structures. The well known results that almost all finite structures are rigid and that the class of finite structures has a zero-one law are, in the present context, the first layer in a hierarchy of classes of finite structures with increasingly more complex automorphism groups. Such a hierarchy can be defined in more than one way. For example, the $k$th level of the hierarchy can consist of all structures having at least $k$ elements which are moved by some automorphism. Or we can consider, for any finite group $G$, all finite structures $\mathcal{M}$ such that $G$ is a subgroup of the group of autmorphisms of $\mathcal{M}$; in this case the "hierarchy" is a partial order. In both cases, as well as variants of them, each "level" satisfies a logical limit law, but not a zero-one law (unless $k = 0$ or $G$ is trivial). Moreover, the number of (labelled or unlabelled) $n$-element structures in one place of the hierarchy divided by the number of $n$-element structures in another place always converges to a rational number or to $\infty$ as $n \to \infty$. All instances of the respective result are proved by an essentially uniform argument.
A condition, in two variants, is given such that if a property P satisfies this condition, then every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The result is used to prove that for a number of natural properties P speaking about automorphism groups, every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The basic idea underlying the results and examples presented here is that, using results from random graph theory, it is possible to construct a countable first-order theory T such that every model of T has a very rich automorphism group, but every finite subset of T has a model which is rigid.
A condition, in two variants, is given such that if a property P satisfies this condition, then every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. The result is used to prove that for a number of natural properties P speaking about automorphism groups or connectivity, every logic which is at least as strong as first-order logic and can express P fails to have the compactness property. We also give an example of a logic that extends first-order logic, has the compactness property and can express the property "the cardinality of the automorphism group is at most $2^{\aleph_0}$".
For integers $l \geq 2$, $d \geq 1$ we study (undirected) graphs with vertices $1, ..., n$ such that the vertices can be partitioned into $l$ parts such that every vertex has at most $d$ neighbours in its own part. The set of all such graphs is denoted $\mbP_n(l,d)$. We prove a labelled first-order limit law, i.e., for every first-order sentence $φ$, the proportion of graphs in $\mbP_n(l,d)$ that satisfy $φ$ converges as $n \to \infty$. By combining this result with a result of Hundack, Prömel and Steger \cite{HPS} we also prove that if $1 \leq s_1 \leq ... \leq s_l$ are integers, then $\mb{Forb}(\mcK_{1, s_1, ..., s_l})$ has a labelled first-order limit law, where $\mb{Forb}(\mcK_{1, s_1, ..., s_l})$ denotes the set of all graphs with vertices $1, ..., n$, for some $n$, in which there is no subgraph isomorphic to the complete $(l+1)$-partite graph with parts of sizes $1, s_1, ..., s_l$. In the course of doing this we also prove that there exists a first-order formula $ξ$ (depending only on $l$ and $d$) such that the proportion of $\mcG \in \mbP_n(l,d)$ with the following property approaches 1 as $n \to \infty$: there is a unique partition of $\{1, ..., n\}$ into $l$ parts such that every vertex has at most $d$ neighbours in its own part, and this partition, viewed as an equivalence relation, is defined by $ξ$.
For any fixed integer $R \geq 2$ we characterise the typical structure of undirected graphs with vertices $1, ..., n$ and maximum degree $R$, as $n$ tends to infinity. The information is used to prove that such graphs satisfy a labelled limit law for first-order logic. If $R \geq 5$ then also an unlabelled limit law holds.
We study finite $l$-colourable structures with an underlying pregeometry. The probability measure that is used corresponds to a process of generating such structures (with a given underlying pregeometry) by which colours are first randomly assigned to all 1-dimensional subspaces and then relationships are assigned in such a way that the colouring conditions are satisfied but apart from this in a random way. We can then ask what the probability is that the resulting structure, where we now forget the specific colouring of the generating process, has a given property. With this measure we get the following results: 1. A zero-one law. 2. The set of sentences with asymptotic probability 1 has an explicit axiomatisation which is presented. 3. There is a formula $ξ(x,y)$ (not directly speaking about colours) such that, with asymptotic probability 1, the relation "there is an $l$-colouring which assigns the same colour to $x$ and $y$" is defined by $ξ(x,y)$. 4. With asymptotic probability 1, an $l$-colourable structure has a unique $l$-colouring (up to permutation of the colours).
We consider a set $\mbK = \bigcup_{n \in \mbbN}\mbK_n$ of {\em finite} structures such that all members of $\mbK_n$ have the same universe, the cardinality of which approaches $\infty$ as $n\to\infty$. Each structure in $\mbK$ may have a nontrivial underlying pregeometry and on each $\mbK_n$ we consider a probability measure, either the uniform measure, or what we call the {\em dimension conditional measure}. The main questions are: What conditions imply that for every extension axiom $φ$, compatible with the defining properties of $\mbK$, the probability that $φ$ is true in a member of $\mbK_n$ approaches 1 as $n \to \infty$? And what conditions imply that this is not the case, possibly in the strong sense that the mentioned probability approaches 0 for some $φ$? If each $\mbK_n$ is the set of structures with universe ${1, ..., n}$, in a fixed relational language, in which certain "forbidden" structures cannot be weakly embedded and $\mbK$ has the disjoint amalgamation property, then there is a condition (concerning the set of forbidden structures) which, if we consider the uniform measure, gives a dichotomy; i.e. the condition holds if and only if the answer to the first question is `yes'. In general, we do not obtain a dichotomy, but we do obtain a condition guaranteeing that the answer is `yes' for the first question, as well as a condition guaranteeing that the answer is `no'; and we give examples showing that in the gap between these conditions the answer may be either `yes' or `no'. This analysis is made for both the uniform measure and for the dimension conditional measure. The later measure has closer relation to random generation of structures and is more "generous" with respect to satisfiability of extension axioms.