Finite subgraphs of uncountably chromatic graphs
Logic
2007-05-23 v1
Abstract
It is consistent that for every monotonically increasing function f:omega->omega there is a graph with size and chromatic number aleph_1 in which every n-chromatic subgraph has at least f(n) elements (n >= 3). This solves a $250 problem of Erdos. It is also consistent that there is a graph X with Chr(X)=|X|= aleph_1 such that if Y is a graph all whose finite subgraphs occur in X then Chr(Y)<=aleph_2 (so the Taylor conjecture may fail).
Cite
@article{arxiv.math/0212064,
title = {Finite subgraphs of uncountably chromatic graphs},
author = {Péter Komjáth and Saharon Shelah},
journal= {arXiv preprint arXiv:math/0212064},
year = {2007}
}