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**1 - 4**of**4**### GENERAL NON-STRUCTURE THEORY

, 2010

"... The theme of the first two sections, is to prepare the framework of how from a “complicated ” family of index models I ∈ K1 we build many and/or complicated structures in a class K2. The index models are characteristically linear orders, trees with κ+1 levels (possibly with linear order on the set o ..."

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The theme of the first two sections, is to prepare the framework of how from a “complicated ” family of index models I ∈ K1 we build many and/or complicated structures in a class K2. The index models are characteristically linear orders, trees with κ+1 levels (possibly with linear order on the set of successors of a member) and linearly ordered graph, for this we phrase relevant complicatedness properties (called bigness). We say when M ∈ K2 is represented in I ∈ K1. We give sufficient conditions when {MI: I ∈ K1 λ} is complicated where for each I ∈ K1 λ we build MI ∈ K2 (usually ∈ K2 λ) represented in it and reflecting to some degree its structure (e.g. for I a linear order we can build a model of an unstable first order class reflecting the order). If we understand enough we can even build e.g. rigid members of K2 λ. Note that we mention “stable”, “superstable”, but in a self contained way, using an equivalent definition which is useful here and explicitly given. We also frame the use of generalizations of Ramsey and Erdös-Rado theorems to

### DEPENDENT THEORIES AND THE GENERIC PAIR CONJECTURE

"... Abstract. On the one hand we try to understand complete types over a somewhat saturated model of a complete first order theory which is dependent, by “decomposition theorems for such types”. Our thesis is that the picture of dependent theory is the combination of the one for stable theories and the ..."

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Abstract. On the one hand we try to understand complete types over a somewhat saturated model of a complete first order theory which is dependent, by “decomposition theorems for such types”. Our thesis is that the picture of dependent theory is the combination of the one for stable theories and the one for the theory of dense linear order or trees (and first we should try to understand the quite saturated case). As a measure of our progress, we give several applications considering some test questions; in particular we try to prove the generic pair conjecture and do it for measurable cardinals.