Abstract Tverberg’s theorem states that a set with sufficiently many points in $${\mathbb {R}}^d$$ R d can always be partitioned into m parts such that the nerve (the intersection pattern) of the convex hulls of the parts form an $$(m-1)$$ ( m - 1 ) -simplex. De Loera, Hogan, Oliveros, and Yang (2021) explored how other simplicial complexes can emerge as nerve complexes for sufficiently large point sets. In this paper, we establish a connection between the theory of word-representable graphs and a method for encoding the 1-skeletons of simplicial complexes to generate nerve complexes. Specifically, we demonstrate that every triangle-free 2-word-representable graph can be realized as a nerve complex in the plane, given sufficiently many points. Furthermore, for every bipartite graph, there exists a dimension d such that it can be represented as a nerve complex for sufficiently many points in $${\mathbb {R}}^d$$ R d .

Abstract. Given a finite set of points in [Formula: see text], Tverberg’s theorem guarantees the existence of partitions of this set into parts whose convex hulls intersect. We introduce a graph structured on the family of Tverberg partitions of a given set of points, whose edges describe closeness between different Tverberg partitions. We prove bounds on the minimum and maximum degrees of this graph, the number of vertices of maximal degree, its clique number, and its connectedness.

Abstract Let S be a set of n points in general position in the plane, and let $$X_{k,\ell }(S)$$ X k , ℓ ( S ) be the number of convex k-gons with vertices in S that have exactly $$\ell $$ ℓ points of S in their interior. We prove several equalities for the numbers $$X_{k,\ell }(S)$$ X k , ℓ ( S ) . This problem is related to the Erdős–Szekeres theorem. Some of the obtained equations also extend known equations for the numbers of empty convex polygons to polygons with interior points. Analogous results for higher dimension are shown as well.