Abstract A structure ${\mathbb Y}$ of a relational language L is called almost chainable iff there are a finite set $F \subset Y$ and a linear order $\,<$ on the set $Y\setminus F$ such that for each partial automorphism $\varphi $ (i.e., local automorphism, in Fraïssé’s terminology) of the linear order $\langle Y\setminus F, <\rangle $ the mapping $\mathop {\mathrm {id}}\nolimits _F \cup \varphi $ is a partial automorphism of ${\mathbb Y}$ . By theorems of Fraïssé and Pouzet, an infinite structure ${\mathbb Y}$ is almost chainable iff the profile of ${\mathbb Y}$ is bounded; namely, iff there is a positive integer m such that ${\mathbb Y}$ has $\leq m$ non-isomorphic substructures of size n, for each positive integer n. A complete first order L-theory ${\mathcal T}$ having infinite models is called almost chainable iff all models of ${\mathcal T}$ are almost chainable and it is shown that the last condition is equivalent to the existence of one countable almost chainable model of ${\mathcal T}$ . In addition, it is proved that an almost chainable theory has either one or continuum many non-isomorphic countable models and, thus, the Vaught conjecture is confirmed for almost chainable theories.

Abstract A relational structure is called reversible iff each bijective endomorphism (condensation) of that structure is an automorphism. We show that reversibility is an invariant of some forms of L∞ω −bi-interpretability, implying that the condensation monoids of structures are topologically isomorphic. Applying these results, we prove that, in particular, all orbits of ultrahomogeneous tournaments and reversible ultrahomogeneous m-uniform hypergraphs are reversible relations and that the same holds for the orbits of reversible ultrahomogeneous digraphs definable by formulas which are not R-negative.

Abstract We investigate the partial orderings of the form P(X),⊂〉, where X is a relational structure and P(X) the set of the domains of its isomorphic substructures. A rough classification of countable binary structures corresponding to the forcing-related properties of the posets of their copies is obtained.

Abstract The game is played on a complete Boolean algebra , by two players. White and Black, in κ-many moves (where κ is an infinite cardinal). At the beginning White chooses a non-zero element p ∈ . In the α-th move White chooses pα ∈ (0, p) and Black responds choosing iα ∈{0, 1}. White winsthe play iff . where and . The corresponding game theoretic properties of c.B.a.'s are investigated. So, Black has a winning strategy (w.s.) if κ ≥ π( ) or if contains a κ-closed dense subset. On the other hand, if White has a w.s., then κ ∈ . The existence of w.s. is characterized in a combinatorial way and in terms of forcing. In particular, if 2<κ = κ ∈ Reg and forcing by preserves the regularity of κ, then White has a w.s. iff the power 2κ is collapsed to κ in some extension. It is shown that, under the GCH, for each set S ⊆ Reg there is a c.B.a. such that White (respectively. Black) has a w.s. for each infinite cardinal κ ∈ S (resp. κ ∉ S). Also it is shown consistent that for each κ ∈ Reg there is a c.B.a. on which the game is undetermined.

Abstract A maximal almost disjoint (mad) family ⊆ [ω]ω is Cohen-stable if and only if it remains maximal in any Cohen generic extension. Otherwise it is Cohen-unstable. It is shown that a mad family. .is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the sets G[A]. A ∈ are nowhere dense. An ℵ0-mad family, . is a mad family with the property that given any countable family ℬ ⊂ [ω]ω such that each element of ℬ meets infinitely many elements of in an infinite set there is an element of meeting each element of ℬ in an infinite set. It is shown that Cohen-stable mad families exist if and only if there exist ℵ0-mad families. Either of the conditions b = c or a < cov( ) implies that there exist Cohen-stable mad families. Similar results are obtained for splitting families. For example, a splitting family. . is Cohen-unstable if and only if there is a bijection G from ω to the rationals such that the boundaries of the sets G[S], S ∈ are nowhere dense. Also. Cohen-stable splitting families of cardinality ≤ κ exist if and only if ℵ0-splitting families of cardinality ≤ κ exist.