arXiv subjects

R. Grossberg

Publications and source records attributed to R. Grossberg.

3 recordsLinked to original sources

arXiv2015

Uniqueness of Limit Models in Classes with Amalgamation

Let K be an abstract elementary class satisfying the joint embedding and the amalgamation properties. Let m be a cardinal above the the Löwenheim-Skolem number of the class. Suppose K satisfies the disjoint amalgamation property for limit models of cardinality m. If K is m-Galois-stable, has no m-Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, for the precise description of long splitting chains and locality}, then any two (m,sigma_i)-limits over M for (i in {1,2}) are isomorphic over M. This theorem extends results of Shelah, Kolman and Shelah, and Shelah and Villaveces. A preliminary version of our uniqueness theorem was used by Grossberg and VanDieren to prove a case of Shelah's categoricity conjecture for tame abstract elementary classes.

math.LO↗

arXiv1998

Transferring saturation, the finite cover property, and stability

Saturation is (mu,kappa)-transferable in T if and only if there is an expansion T_1 of T with |T_1| = |T| such that if M is a mu-saturated model of T_1 and |M| \geq kappa then the reduct M|L(T) is kappa-saturated. We characterize theories which are superstable without the finite cover property (f.c.p.), or without f.c.p. as, respectively those where saturation is (aleph_0,lambda)-transferable or (kappa(T),lambda)-transferable for all lambda. Further if for some mu \geq |T|, 2^mu > mu^+, stability is equivalent to: or all mu \geq |T|, saturation is (μ,2^mu)-transferable.

math.LO↗

arXiv1994

Infinite homogeneous bipartite graphs with unequal sides

We call a bipartite graph {\it homogeneous} if every finite partial automorphism which respects left and right can be extended to a total automorphism. A $(κ,λ )$ bipartite graph is a bipartite graph with left side of size $κ$ and right side of size $λ$. We show, using a theorem of Hrushovski on finite graphs, that there is a homogeneous $({\aleph_0},2^{\aleph_0} )$ bipartite graph of girth 4 (thus answering negatively a question by Kupitz and Perles), and that depending on the underlying set theory all homogeneous $({\aleph_0},\aleph_1)$ bipartite graphs may be isomorphic, or there may be $2^{\aleph_1}$ many isomorphism types of $(\aleph_0,\aleph_1)$ homogeneous graphs.

math.LO↗