Let $\mathscr {C}$ be a class of finite and infinite graphs that is closed under induced subgraphs. The well-known Łoś–Tarski Theorem from classical model theory implies that $\mathscr {C}$ is definable in first-order logic by a sentence $\varphi $ if and only if $\mathscr {C}$ has a finite set of forbidden induced finite subgraphs. This result provides a powerful tool to show nontrivial characterizations of graphs of small vertex cover, of bounded tree-depth, of bounded shrub-depth, etc. in terms of forbidden (...) induced finite subgraphs. Furthermore, by the Completeness Theorem, we can compute from $\varphi $ the corresponding forbidden induced subgraphs. This machinery fails on finite graphs as shown by our results: –There is a class $\mathscr {C}$ of finite graphs that is definable in first-order logic and closed under induced subgraphs but has no finite set of forbidden induced subgraphs.–Even if we only consider classes $\mathscr {C}$ of finite graphs that can be characterized by a finite set of forbidden induced subgraphs, such a characterization cannot be computed from a first-order sentence $\varphi $ that defines $\mathscr {C}$ and the size of the characterization cannot be bounded by $f(|\varphi |)$ for any computable function f.Besides their importance in graph theory, the above results also significantly strengthen similar known theorems for arbitrary structures. (shrink)

We answer some questions about graphs that are reducts of countable models of Anti-Foundation, obtained by considering the binary relation of double-membership $x\in y\in x$. We show that there are continuum-many such graphs, and study their connected components. We describe their complete theories and prove that each has continuum-many countable models, some of which are not reducts of models of Anti-Foundation.

By a celebrated theorem of Morley, a theory T is ℵ1‐categorical if and only if it is κ‐categorical for all uncountable κ. In this paper we are taking the first steps towards extending Morley's categoricity theorem “to the finite”. In more detail, we are presenting conditions, implying that certain finite subsets of certain ℵ1‐categorical T have at most one n‐element model for each natural number (counting up to isomorphism, of course).

We investigate degree of satisfiability questions in the context of Heyting algebras and intuitionistic logic. We classify all equations in one free variable with respect to finite satisfiability gap, and determine which common principles of classical logic in multiple free variables have finite satisfiability gap. In particular we prove that, in a finite non-Boolean Heyting algebra, the probability that a randomly chosen element satisfies $x \vee \neg x = \top $ is no larger than $\frac {2}{3}$. Finally, we generalize our (...) results to infinite Heyting algebras, and present their applications to point-set topology, black-box algebras, and the philosophy of logic. (shrink)

We will show that almost all nonassociative relation algebras are symmetric and integral (in the sense that the fraction of both labelled and unlabelled structures that are symmetric and integral tends to $1$ ), and using a Fraïssé limit, we will establish that the classes of all atom structures of nonassociative relation algebras and relation algebras both have $0$ – $1$ laws. As a consequence, we obtain improved asymptotic formulas for the numbers of these structures and broaden some known probabilistic (...) results on relation algebras. (shrink)