Semialgebraic Graphs having Countable List-Chromatic Numbers

The set of semialgebraic graphs having countable list-chromatic numbers is characterized. Some other related sets of graphs having countable list-chromatic numbers also are.Comment: This version has been completely rewritten. It will appear in PAM

The Pentagon Lattice as a Substructure Lattice of Models of Peano Arithmetic

If $M \prec N$ are models of Peano Arithmetic and Lt$(N/M)$ is the pentagon lattice $N_5$, then $N$ is either a cofinal or an end extension of $M$. In contrast, there are $M \prec N$ that are models of PA* (PA in a language with countably many new predicate symbols) such that Lt$(N/M) \cong N_5$ and $N$ is neither a cofinal nor an end extension of $M$.Comment: This paper is being withdrawn due to a serious erro