Embedding Dimension Phenomena in Intersection Complete Codes

Abstract

Two tantalizing invariants of a combinatorial code $\mathcal C\subseteq 2^{[n]}$ are cdim$(\mathcal C)$ and odim$(\mathcal C)$, the smallest dimension in which $\mathcal C$ can be realized by convex closed or open sets, respectively. Cruz, Giusti, Itskov, and Kronholm showed that for intersection complete codes $\mathcal C$ with $m+1$ maximal codewords, odim$(\mathcal C)$ and cdim$(\mathcal C)$ are both bounded above by $\max\{2,m\}$. Results of Lienkaemper, Shiu, and Woodstock imply that odim and cdim may differ, even for intersection complete codes. We add to the literature on open and closed embedding dimensions of intersection complete codes with the following results: (*) If $\mathcal C$ is a simplicial complex, then cdim$(\mathcal{C}) = \mbox{odim}(\mathcal C)$, (*) If $\mathcal C$ is intersection complete, then cdim$(\mathcal C)\le \mbox{odim}(\mathcal C)$, (*) If $\mathcal C\subseteq 2^{[n]}$ is intersection complete with $n\ge 2$, then cdim$(\mathcal C) \le \min \{2d+1, n-1\}$, where $d$ is the dimension of the simplicial complex of $\mathcal C$, and (*) For each simplicial complex $\Delta\subseteq 2^{[n]}$ with $m\ge 2$ facets, the code $\mathcal S_\Delta := (\Delta \ast (n+1)) \cup \{[n]\}\subseteq 2^{[n+1]}$ is intersection complete, has $m+1$ maximal codewords, and satisfies odim$(\mathcal S_\Delta)=m$. In particular, for each $n\ge 3$ there exists an intersection complete code $\mathcal C\subseteq 2^{[n]}$ with odim$(\mathcal C) = \binom{n-1}{\lfloor (n-1)/2\rfloor}$. A key tool in our work is the study of sunflowers: arrangements of convex open sets in which the sets simultaneously meet in a central region, and nowhere else. We use Tverberg's theorem to study the structure of "$k$-flexible" sunflowers, and consequently obtain new lower bounds on odim$(\mathcal C)$ for intersection complete codes $\mathcal C$.