On incompactness for chromatic number of graphs
  We deal with incompactness. Assume the existence of non-reflecting stationary set of cofinality kappa . We prove that one can define a graph G whose chromatic number is > kappa, while the chromatic number of every subgraph G' subseteq G,|G'| < |G| is <= kappa . The main case is kappa = aleph_0. 
Saharon Shelah
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