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 N₁ in which every n-chromatic subgraph contains at least f(n) vertices (n ≥ 3). This solves a $ 250 problem of Erdo″s. It is consistent that there is a graph X with Chr(X) = |X| = N₁ such that if Y is a graph all whose finite subgraphs occur in X then Chr(Y) ≤ N₂ (so the Taylor conjecture may fail). It is also consistent that if X is a graph with chromatic number at least N₂ then for every cardinal λ there exists a graph Y with Chr(V) ≥ λ all whose finite subgraphs are induced subgraphs of X.