  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).
