Intersection patterns of finite sets and of convex sets

TL;DR

This paper unifies and extends key results in hypergraph chromatic numbers and Tverberg-type intersection theorems, providing elementary proofs and new bounds through geometric and topological methods.

Contribution

It offers a new generalization linking hypergraph coloring bounds with intersection patterns in simplicial complexes, simplifying proofs of classical results.

Findings

Elementary proof of Kneser's conjecture

Lower bounds for chromatic numbers of Kneser hypergraphs

Connections between equivariant maps and coloring bounds

Abstract

The main result is a common generalization of results on lower bounds for the chromatic number of r-uniform hypergraphs and some of the major theorems in Tverberg-type theory, which is concerned with the intersection pattern of faces in a simplicial complex when continuously mapped to Euclidean space. As an application we get a simple proof of a generalization of a result of Kriz for certain parameters. This specializes to a short and simple proof of Kneser's conjecture. Moreover, combining this result with recent work of Mabillard and Wagner we show that the existence of certain equivariant maps yields lower bounds for chromatic numbers. We obtain an essentially elementary proof of the result of Schrijver on the chromatic number of stable Kneser graphs. In fact, we show that every neighborly even-dimensional polytope yields a small induced subgraph of the Kneser graph of the same chromatic number. We furthermore use this geometric viewpoint to give tight lower bounds for the chromatic number of certain small subhypergraphs of Kneser hypergraphs.