Multiple recurrence of Markov shifts and other infinite measure preservingtransformations

We discuss the concept of multiple recurrence, considering an ergodic versi on of a conjecture of Erdos. This conjecture applies to infinite measure pr eserving transformations. We prove a result stronger than the ergodic conje cture for the class of Markov shifts and show by example that our stronger result is not true for all measure preserving transformations.