For every natural number r, the set of r-tuples of natural numbers is partitioned into finitely many classes. Show that if f(r) is a function such that f(r)≥1 and \lim _{r\rightarrow \infty} f(r)\equal{}\plus{}\infty, then there exists an infinite set of natural numbers that, for all r, contains r-triples from at most f(r) classes. Show that if f(r) \not \rightarrow \plus{}\infty, then there is a family of partitions such that no such infinite set exists.
P. Erdos, A. Hajnal functionlimitadvanced fieldsadvanced fields unsolved