Generic graph construction.

*(English)*Zbl 0573.03021P. 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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