  Convex neural codes are subsets of the Boolean lattice that record the
intersection patterns of convex sets in Euclidean space. Much work in recent
years has focused on finding combinatorial criteria on codes that can be used
to classify whether or not a code is convex. In this paper we introduce
order-forcing, a combinatorial tool which recognizes when certain regions in a
realization of a code must appear along a line segment between other regions.
We use order-forcing to construct novel examples of non-convex codes, and to
expand existing families of examples. We also construct a family of codes which
shows that a dimension bound of Cruz, Giusti, Itskov, and Kronholm (referred to
as monotonicity of open convexity) is tight in all dimensions.
