Finite subgraphs of uncountably chromatic graphs

by Komjath and Shelah. [KoSh:788]
J Graph Theory., 2005
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 textdollar ~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 .

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