More on Compactness of Chromatic Numbers

TL;DR

This paper constructs graphs with high chromatic number and specific size properties, demonstrating the relationship between graph size and chromatic number in the context of infinite cardinals.

Contribution

It introduces a method to build graphs with prescribed chromatic numbers and sizes, extending understanding of chromatic properties in infinite graph theory.

Findings

Existence of graphs with chromatic number > kappa and size mu^kappa

Subgraphs of smaller size have bounded chromatic number

Results hold below the first fixed point of aleph_lambda

Abstract

We prove that for any regular kappa and mu > kappa below the first fix point (lambda = aleph_lambda) above kappa, there is a graph with chromatic number > kappa, and mu^kappa nodes but every subgraph of cardinality < mu has chromatic number less than or equal to kappa.