Convex Hull of Face Vectors of Colored Complexes

TL;DR

This paper proves a conjecture about the convex hull of face vectors in r-colorable complexes, extending Turán's theorem, and provides insights into the combinatorial structure of colored complexes.

Contribution

It verifies Kozlov's conjecture and generalizes Turán's theorem within the context of face vectors of colored complexes.

Findings

Confirmed Kozlov's conjecture on convex hulls of face vectors.

Derived a generalized version of Turán's graph theorem.

Enhanced understanding of combinatorial properties of colored complexes.

Abstract

In this paper we verify a conjecture by Kozlov (Discrete Comput Geom 18 (1997) 421--431), which describes the convex hull of the set of face vectors of $r$-colorable complexes on $n$ vertices. As part of the proof we derive a generalization of Tur\'{a}n's graph theorem.