Many symmetrically indivisible structures

Abstract

A structure $\mathcal{M}$ in a first-order language $\mathcal{L}$ is \emph{indivisible} if for every coloring of $M$ in two colors, there is a monochromatic $\mathcal{M}^{\prime} \subseteq \mathcal{M}$ such that $\mathcal{M}^{\prime}\cong\mathcal{M}$. Additionally, we say that $\mathcal{M}$ is symmetrically indivisible if $\mathcal{M}^{\prime}$ can be chosen to be \emph{symmetrically embedded} in $\mathcal{M}$ (that is, every automorphism of $\mathcal{M}^{\prime}$ can be extended to an automorphism of $\mathcal{M}$). In the following paper we give a general method for constructing new symmetrically indivisible structures out of existing ones. Using this method, we construct $2^{\aleph_0}$ many non-isomorphic symmetrically indivisible countable structures in given (elementary) classes and answer negatively the following question asked by A. Hasson, M. Kojman and A. Onshuus in "On symmetric indivisibility of countable structures" (Cont. Math. 558(1):453--466): Let $\mathcal{M}$ be a symmetrically indivisible structure in a language $\mathcal{L}$. Let $\mathcal{L}_0 \subseteq \mathcal{L}$. Is $ \mathcal{M} \upharpoonright \mathcal{L}_0$ symmetrically indivisible?