Abstract A permutation group G on a set A is ${\kappa }$ -homogeneous iff for all $X,Y\in \bigl [ {A} \bigr ]^ {\kappa } $ with $|A\setminus X|=|A\setminus Y|=|A|$ there is a $g\in G$ with $g[X]=Y$ . G is ${\kappa }$ -transitive iff for any injective function f with $\operatorname {dom}(f)\cup \operatorname {ran}(f)\in \bigl [ {A} \bigr ]^ {\le {\kappa }} $ and $|A\setminus \operatorname {dom}(f)|=|A\setminus \operatorname {ran}(f)|=|A|$ there is a $g\in G$ with $f\subset g$ . Giving a partial answer to a question of P. M. Neumann [6] we show that there is an ${\omega }$ -homogeneous but not ${\omega }$ -transitive permutation group on a cardinal ${\lambda }$ provided (i) ${\lambda }<{\omega }_{\omega }$ , or (ii) $2^{\omega }<{\lambda }$ , and ${\mu }^{\omega }={\mu }^+$ and $\Box _{\mu }$ hold for each ${\mu }\le {\lambda }$ with ${\omega }=\operatorname {cf}({\mu })<{{\mu }}$ , or (iii) our model was obtained by adding $(2^{\omega })^+$ many Cohen generic reals to some ground model. For ${\kappa }>{\omega }$ we give a method to construct large ${\kappa }$ -homogeneous, but not ${\kappa }$ -transitive permutation groups. Using this method we show that there exist ${\kappa }^+$ -homogeneous, but not ${\kappa }^+$ -transitive permutation groups on ${\kappa }^{+n}$ for each infinite cardinal ${\kappa }$ and natural number $n\ge 1$ provided $V=L$ .

For any regular cardinal κ and ordinal η<κ++ it is consistent that 2κ is as large as you wish, and every function f:η⟶[κ,2κ]∩Card with f(α)=κ for cf(α)<κ is the cardinal sequence of some locally compact scattered space.

Abstract We explore a general method based on trees of elementary submodels in order to present highly simplified proofs to numerous results in infinite combinatorics. While countable elementary submodels have been employed in such settings already, we significantly broaden this framework by developing the corresponding technique for countably closed models of size continuum. The applications range from various theorems on paradoxical decompositions of the plane, to coloring sparse set systems, results on graph chromatic number and constructions from point-set topology. Our main purpose is to demonstrate the ease and wide applicability of this method in a form accessible to anyone with a basic background in set theory and logic.

We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side by-side product of partial orderings which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with ¬CH and Martin's axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the club principle for every stationary subset of limits of ω1 together with ¬CH and Martin's axiom for countable p.o.-sets.