Chromatic Numbers of Cayley Graphs on Z and Recurrence

Dedicated to the memory of Paul Erdős

In 1987 Paul Erdős asked me if the Cayley graph defined on Z by a lacunary sequence has necessarily a finite chromatic number. Below is my answer, delivered to him on the spot but never published, and some additional remarks. The key is the interpretation of the question in terms of return times of dynamical systems.

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Received February 7, 2000

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Katznelson, Y. Chromatic Numbers of Cayley Graphs on Z and Recurrence. Combinatorica 21, 211–219 (2001).

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  • AMS Subject Classification (2000) Classes:  05C15, 37B20