Abstract
AbstractIt is consistent that for every function f:ω → ω there is a graph with size and chromatic number ℵ1 in which every n‐chromatic subgraph contains at least f(n) vertices (n ≥ 3). This solves a $ 250 problem of Erdős. It is consistent that there is a graph X with Chr(X)=|X|=ℵ1 such that if Y is a graph all whose finite subgraphs occur in X then Chr(Y)≤ℵ2 (so the Taylor conjecture may fail). It is also consistent that if X is a graph with chromatic number at least ℵ2 then for every cardinal λ there exists a graph Y with Chr(Y)≥λ all whose finite subgraphs are induced subgraphs of X. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 28–38, 2005
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