Finite subgraphs of uncountably chromatic graphs

It 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.