From word-representable graphs to altered Tverberg-type theorems

Abstract

Tverberg's theorem says that a set with sufficiently many points in $\mathbb{R}^d$ can always be partitioned into $m$ parts so that the $(m-1)$-simplex is the (nerve) intersection pattern of the convex hulls of the parts. In arXiv:1808.00551v1 [math.MG] the authors investigate how other simplicial complexes arise as nerve complexes once we have a set with sufficiently many points. In this paper we relate the theory of word-representable graphs as a way of codifying $1$-skeletons of simplicial complexes to generate nerves. In particular, we show that every $2$-word-representable triangle-free graph, every circle graph, every outerplanar graph, and every bipartite graph could be induced as a nerve complex once we have a set with sufficiently many points in $\mathbb{R}^d$ for some $d$.