Chromatic Numbers of Cayley Graphs on Z and Recurrence

  title={Chromatic Numbers of Cayley Graphs on Z and Recurrence},
  author={Y. Katznelson},
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 never1published, and some additional remarks. The key is the interpretation of the question in terms of return times of dynamical systems. 1 The Cayley graph ZΛ 1.1 Let Λ ⊂ N. By definition, the Cayley graph ZΛ is the graph whose vertices are the integers, and whose edges are the pairs {(n, n+ λ) : n ∈ Z, λ ∈ Λ}. The… CONTINUE READING