124,706 research outputs found
Binary simple homogeneous structures are supersimple with finite rank
Suppose that M is an infinite structure with finite relational vocabulary
such that every relation symbol has arity at most 2. If M is simple and
homogeneous then its complete theory is supersimple with finite SU-rank which
cannot exceed the number of complete 2-types over the empty set
A limit law of almost -partite graphs
For integers , we study (undirected) graphs with
vertices such that the vertices can be partitioned into parts
such that every vertex has at most neighbours in its own part. The set of
all such graphs is denoted \mbP_n(l,d). We prove a labelled first-order limit
law, i.e., for every first-order sentence , the proportion of graphs
in \mbP_n(l,d) that satisfy converges as . By
combining this result with a result of Hundack, Pr\"omel and Steger \cite{HPS}
we also prove that if are integers, then
\mb{Forb}(\mcK_{1, s_1, ..., s_l}) has a labelled first-order limit law,
where \mb{Forb}(\mcK_{1, s_1, ..., s_l}) denotes the set of all graphs with
vertices , for some , in which there is no subgraph isomorphic to
the complete -partite graph with parts of sizes . In
the course of doing this we also prove that there exists a first-order formula
(depending only on and ) such that the proportion of \mcG \in
\mbP_n(l,d) with the following property approaches 1 as : there
is a unique partition of into parts such that every vertex
has at most neighbours in its own part, and this partition, viewed as an
equivalence relation, is defined by
- …