Let T be a simple L-theory and let \ be a reduct of T to a sublanguage \ of L. For variables x, we call an \-invariant set \\) in \ a universal transducer if for every formula \\in L^-\) and every a, $$\begin{aligned} \phi ^-\ L^-\text{-forks } \text{ over }\ \emptyset \ \text{ iff } \Gamma \wedge \phi ^-\ L\text{-forks } \text{ over }\ \emptyset. \end{aligned}$$We show that there is a greatest universal transducer \ and it is type-definable. In (...) particular, the forking topology on \\) refines the forking topology on \\) for all y. Moreover, we describe the set of universal transducers in terms of certain topology on the Stone space and show that \ is the unique universal transducer that is \-type-definable with parameters. If \ is a theory with the wnfcp and T is the theory of its lovely pairs of models we show that \\) and give a more precise description of the set of universal transducers for the special case where \ has the nfcp. (shrink)

The binding group theorem for stable theories is partially extended to the simple context. Some results concerning internality are proved. We also introduce a ‘small’ normal subgroup G0+ of the automorphism group and show that if p is Q-internal then it has a finite exponent and G/G0+ is interpretable.

We show that in a simple theory T in which the τf-topologies are closed under projections every type analyzable in a supersimple τf-open set has ordinal SU-rank. In particular, if in addition T is unidimensional, the existence of a supersimple unbounded τf-open set implies T is supersimple. We also introduce the notion of a standard τ-metric and show that for simple theories its completeness is equivalent to the compactness of the τ-topology.

In a simple theory with elimination of finitary hyperimaginaries if tp(a) is real and analysable over a definable set Q, then there exists a finite sequence ( $a_{i}|i \leq n^{*}$ ) $\subseteq dcl^{eq}$ (a) with $a_{n}*$ = a such that for every $i \leq n*$ , if $p_{i} = tp(a_{i}/{a_{i}|j < i}$ ) then $Aut(p_{i}/Q)$ is type-definable with its action on $p_{i}^{c}$ . A unidimensional simple theory eliminates the quantifier $\exists^{\infty}$ and either interprets (in $C^{eq}$ ) an infinite type-definable group (...) or has the property that ACL(Q) = C for every infinite definable set Q. (shrink)

We extend the dichotomy between 1-basedness and supersimplicity proved in Shami :309–332, 2011). The generalization we get is to arbitrary language, with no restrictions on the topology [we do not demand type-definabilty of the open set in the definition of essential 1-basedness from Shami :309–332, 2011)]. We conclude that every hypersimple unidimensional theory that is not s-essentially 1-based by means of the forking topology is supersimple. We also obtain a strong version of the above dichotomy in the case where the (...) language is countable. (shrink)

We show that a Kueker simple theory eliminates Ǝ∞ and densely interprets weakly minimal formulas. As part of the proof we generalize Hrushovski's dichotomy for almost complete formulas to simple theories. We conclude that in a unidimensional simple theory an almost-complete formula is either weakly minimal or trivially-almost-complete. We also observe that a small unidimensional simple theory is supersimple of finite SU-rank.

Let T be simple, work in Ceq over a boundedly closed set. Let p ∈ S(θ) be internal in a quasi-stably-embedded type-definable set Q (e.g., Q is definable or stably-embedded) and suppose (p, Q) is ACL-embedded in Q (see definitions below). Then Aut(p/Q) with its action on pC is type-definable in Ceq over θ. In particular, if p ∈ S(θ) is internal in a stably-embedded type-definable set Q, and pC υ Q is stably-embedded, then Aut(p/Q) is type-definable with its action (...) on pC. (shrink)

We generalize Shelah's analysis of cardinality quantifiers for a superstable theory from Chapter V of Classification Theory and the Number of Nonisomorphic Models. We start with a set of bounds for the cardinality of each formula in some general invariant family of formulas in a superstable theory (in Classification Theory, a uniform family of formulas is considered) and find a set of derived bounds for all formulas. The set of derived bounds is sharp: up to a technical restriction, every model (...) that satisfies the original bounds has a sufficiently saturated elementary extension that satisfies the original bounds and such that for each formula the set of its realizations in the extension has arbitrarily large cardinality below the corresponding derived bound of the formula. (shrink)

We point out some connections between existence of homogenous sets for certain edge colorings and existence of branches in certain trees. As a consequence, we get that any locally additive coloring (a notion introduced in the paper) of a cardinal κ has a homogeneous set of size κ provided that the number of colors μ satisfies μ + < κ $\mu ^+<\kappa$. Another result is that an uncountable cardinal κ is weakly compact if and only if κ is regular, has (...) the tree property, and for each λ, μ < κ $\lambda,\mu <\kappa$ there exists κ ∗ < κ $\kappa ^*<\kappa$ such that every tree of height μ with λ nodes has less than κ ∗ $\kappa ^*$ branches. (shrink)

We show that if p is a real type which is almost internal in a formula φ in a simple theory, then there is a type p' interalgebraic with a finite tuple of realizations of p, which is generated over φ. Moreover, the group of elementary permutations of p' over all realizations of φ is type-definable.