Let k > 0. A set R of natural numbers is called a set of measurable k-recurrence if for every measure preserving system (X,T) and every positive measure set A, there exists some n in R and some x in X such that {x,T^nx,...,T^(kn)x} is contained in A. R is called a set of topological k-recurrence if the same conclusion holds for every dynamical system (X,T), where X is compact metric and T is a minimal homeomorphism, and every non-empty open set A.

These two notions of recurrence are related to density Ramsey theory and chromatic Ramsey theory, respectively. One can easily show that any set of measurable k-recurrence is a set of topological k-recurrence. In 1981 H. Furstenberg gave an example of a set of measurable 1-recurrence that is not a set of topological 2-recurrence, and in 1987 I. Kriz gave an example of a set of topological 1-recurrence that is not a set of measurable 1-recurrence. Other matters of this sort, both in the natural numbers and in more general semigroups, remain unresolved.