SCHWEITZER 1968 Problem 7

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 $f(r)\geq 1$ and $\lim _{r\rightarrow \infty} f(r)=+\infty$ , 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 $f(r) \not \rightarrow +\infty$ , then there is a family of partitions such that no such infinite set exists.

P. Erdos, A. Hajnal