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Generic graph construction. (English) Zbl 0573.03021
P. Erdős and A. Hajnal [Theory of Graphs, Proc. Colloquium Tihany, Hungary 1966, 83-98 (1968; Zbl 0164.248)], assuming GCH, constructed an example of a graph G on $$\aleph_ 2$$ vertices having chromatic number $$\aleph_ 1$$ with the property that every subgraph of G on at most $$\aleph_ 1$$ vertices has chromatic number at most $$\aleph_ 0$$. They asked whether there is a graph G on $$\aleph_ 2$$ vertices but having chromatic number $$\aleph_ 2$$ with the same subgraph property. No progress was made on this question until the paper under review. The paper is devoted to showing that the existence of such a graph is relatively consistent with ZF. (The author remarks that Foreman and Laver have shown the consistency of the non-existence of such a graph, using a huge cardinal.) The relative consistency is shown by constructing a forcing model; the forcing conditions are quite complicated. However, the author spends some time indicating why simpler conditions are unlikely to work; this aids considerably in absorbing the details.
Reviewer: N.H.Williams

##### MSC:
 03E35 Consistency and independence results 03E05 Other combinatorial set theory 05C15 Coloring of graphs and hypergraphs
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##### References:
 [1] Infinite and finite sets I pp 403– (1975) [2] Combinatorial set theory (1977) · Zbl 0362.04008
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