On incompactness for chromatic number of graphs

TL;DR

This paper demonstrates the incompactness phenomenon in graph theory, showing that a graph can have a large chromatic number while all its smaller subgraphs have a smaller chromatic number, under certain set-theoretic assumptions.

Contribution

It constructs graphs with incompactness properties related to chromatic number using set-theoretic principles like non-reflecting stationary sets.

Findings

Existence of graphs with chromatic number greater than κ despite all smaller subgraphs having chromatic number ≤ κ.

Application of set-theoretic assumptions to graph coloring problems.

Main case analyzed is κ = ℵ₀.

Abstract

We deal with incompactness. Assume the existence of non-reflecting stationary set of cofinality kappa . We prove that one can define a graph G whose chromatic number is > kappa, while the chromatic number of every subgraph G' subseteq G,|G'| < |G| is <= kappa . The main case is kappa = aleph_0.