  Colour the edges of the complete graph with vertex set $\{1, 2, \dotsc, n\}$
with an arbitrary number of colours. What is the smallest integer $f(l,k)$ such
that if $n > f(l,k)$ then there must exist a monotone monochromatic path of
length $l$ or a monotone rainbow path of length $k$? Lefmann, R\"{o}dl, and
Thomas conjectured in 1992 that $f(l, k) = l^{k - 1}$ and proved this for $l
\ge (3 k)^{2 k}$. We prove the conjecture for $l \geq k^3 (\log k)^{1 + o(1)}$
and establish the general upper bound $f(l, k) \leq k (\log k)^{1 + o(1)} \cdot
l^{k - 1}$. This reduces the gap between the best lower and upper bounds from
exponential to polynomial in $k$. We also generalise some of these results to
the tournament setting.
