d-Representability of simplicial complexes of fixed dimension

TL;DR

This paper proves Wegner's conjecture that there exist d-dimensional simplicial complexes that are not 2d-representable, clarifying the limitations of representing complexes with convex sets in Euclidean space.

Contribution

The paper confirms Wegner's long-standing conjecture that not all d-dimensional complexes are 2d-representable, establishing a fundamental limit in convex set intersection representations.

Findings

Wegner's bound of 2d-representability is tight.

Existence of d-dimensional complexes not 2d-representable.

Advances understanding of convex set intersection patterns.

Abstract

Let K be a simplicial complex with vertex set V = {v_1,..., v_n}. The complex K is d-representable if there is a collection {C_1,...,C_n} of convex sets in R^d such that a subcollection {C_{i_1},...,C_{i_j}} has a nonempty intersection if and only if {v_{i_1},...,v_{i_j}} is a face of K. In 1967 Wegner proved that every simplicial complex of dimension d is (2d+1)-representable. He also suggested that his bound is the best possible, i.e., that there are $d$-dimensional simplicial complexes which are not 2d-representable. However, he was not able to prove his suggestion. We prove that his suggestion was indeed right. Thus we add another piece to the puzzle of intersection patterns of convex sets in Euclidean space.