Abstract 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.

Abstract. Any [Formula: see text]-tuple of points in the plane can be moved to any other [Formula: see text]-tuple by a continuous motion with at most [Formula: see text] intermediate changes of the order type. Even for tuples with the same order type, the cubic bound is sharp: there exist pairs of [Formula: see text]-tuples of the same order type requiring [Formula: see text] intermediate changes. We show that the upper bound can be slightly improved if the order types of the two [Formula: see text]-tuples are either not too elongated, or are not mirror images of one another.

Abstract We study the open, closed, and non-degenerate embedding dimensions of neural codes, which are the smallest respective dimensions in which one can find a realization of a code consisting of convex sets that are open, closed, or non-degenerate in a sense defined by Cruz, Giusti, Itskov, and Kronholm. For a given code $$\mathcal {C}$$ C we define the embedding dimension vector to be the triple (a, b, c) consisting of these embedding dimensions. Existing results guarantee that $$\max {\{a,b\}}\le c$$ max { a , b } ≤ c , and we show that when any of these dimensions is at least 2 this is the only restriction on such vectors. Specifically, for every triple (a, b, c) with $$2\le \min {\{a,b\}}$$ 2 ≤ min { a , b } and $$\max {\{a,b\}}\le c\le \infty $$ max { a , b } ≤ c ≤ ∞ we construct a code $$\mathcal {C}_{(a,b,c)}$$ C ( a , b , c ) whose embedding dimension vector is exactly (a, b, c) (where an embedding dimension is $$\infty $$ ∞ if there is no realization of the corresponding type). Our constructions combine two existing tools in the convex neural codes literature: sunflowers of convex open sets, and rigid structures, the latter of which was recently defined in work of Chan, Johnston, Lent, Ruys de Perez, and Shiu. Our constructions provide the first examples of codes whose closed embedding dimension is larger than their open embedding dimension, but still finite.

Abstract. We show that every convex code realizable by compact sets in the plane admits a realization consisting of polygons, and analogously every open convex code in the plane can be realized by interiors of polygons. We give factorial-type bounds on the number of vertices needed to form such realizations. Consequently we show that there is an algorithm to decide whether a convex code admits a closed or open realization in the plane.