Problem 190

Erdős Problem #190: declared status 'solved'. Formalized: no. Let $H(k)$ be the smallest $N$ such that in any finite colouring of $\{1,\ldots,N\}$ (into any number of colours) there is always either a monochromatic $k$-term arithmetic progression or a rainbow arithmetic progression (i.e. all elements are different colours). Estimate $H(k)$. Is it true that$$H(k)^{1/k}/k \to \infty$$as $k\to\infty$? Prize: no. Tags: additive combinatorics, arithmetic progressions.