Compactness problems for chromatic numbers of graphs

I'll speak about compactness problems for chromatic numbers of graphs. The main result will be a some what simplified proof of a theorem by Shelah, that a non reflecting stationary subset of a regular cardinal \lambda S\subseteq S^\lambda_kappa implies (under mild cardinal arithmetic assumptions) that there is a graph of size \lambda with chromatic number lambda , but every smaller cardinality subgraph has chromatic number <=kappa.