Neural codes with three maximal codewords: convexity and minimal embedding dimension

Neural codes, represented as collections of binary strings called codewords, are used to encode neural activity. A code is convex if its codewords an arrangement open sets in Euclidean space. Previous work has focused on addressing the question: how can we tell when a convex? Giusti and Itskov identified local obstruction proved that codes have no obstructions. The converse true for up four neurons, but false general. Nevertheless, prove this holds with three maximal moreover minimal embedding dimension such at most two.