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Erdős and Hajnal conjectured in 1966 that every graph of uncountable chromatic number contains a subgraph of infinite connectivity. We prove that every graph of uncountable chromatic number has a subgraph which has uncountable chromatic number and infinite edge-connectivity. We also prove that, if each orientation of a graph G has a vertex of infinite outdegree, then G contains an uncountable subgraph of infinite edge-connectivity.

Infinitely connected subgraphs in graphs of uncountable chromatic number