Definition 0.1 Let δ be a cardinal. We say that δ is Woodin with 3 iff there is some sequence (a κ : κ < δ) such that a κ ⊂ V κ for every κ < δ and for every A ⊂ V δ the set {κ < δ : A ∩ V κ = a κ ∧ κ is A–strong up to δ} is stationary in δ. Proof. Let us define ((a κ , c κ) : κ < δ) recursively as follows. If ((a κ , c κ) : κ < µ) is defined for some µ < δ, then we let (a µ , c µ) be the least...

the date of receipt and acceptance should be inserted later Keywords Saturated models · institutions · institution-independent model theory Abstract Saturated models constitute one of the powerful methods of conventional model theory, with many applications. Here we develop a categorical abstract model theoretic approach to saturated models within the theory of institutions. The most important ...

A simple topological graph G is a graph drawn in the plane so that any pair of edges have at most one point in common, which is either an endpoint or a proper crossing. G is called saturated if no further edge can be added without violating this condition. We construct saturated simple topological graphs with n vertices and O(n) edges. These constructions are nearly optimal: it is shown that ev...

A simplified Hückel-type molecular-orbital (MO) model for the valence electrons of saturated hydrocarbons is proposed and the consequent eigenvalue spectrum considered. A first foundational result is obtained, which every chemist “knows”, namely that: alkanes are stable, with half their (Hückeltype MO) eigenvalues positive and half negative.

Let H be a graph. A graph G is said to be H -free if it contains no subgraph isomorphic to H . A graph G is said to be an H -saturated subgraph of a graph K if G is an H -free subgraph of K with the property that for any edge e∈E(K) \ E(G); G ∪ {e} is not H -free. We present some general results on Ks; t-saturated subgraphs of the complete bipartite graph Km;n and study the problem of 3nding, f...

A graph G is triangle-saturated if every possible edge addition to G creates one or more new triangles (3-cycles). Such a graph is minimally triangle-saturated if removal of any edge from G leaves a graph that is not triangle-saturated. This paper investigates adding a single new vertex to a triangle-saturated graph so as to produce a new trianglesaturated graph, and determines the conditions u...

We are proving the following: (1) If κ is a weakly inaccessible then NSκ is not κ -saturated. (2) If κ is a weakly inaccessible and θ < κ is regular then NS κ is not κ saturated. (3) If κ is singular then NS κ+ is not κ-saturated. Combining this with previous results of Shelah, one obtains the following: (A) If κ > א1 then NSκ is not κ -saturated. (B) If θ < κ then NS κ is not κ -saturated.