Flashes and Rainbows in Tournaments

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, Rödl, 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 \(l \ge k^3 (\log k)^{1 + o(1)}\) and establish the general upper bound \(f(l, k) \le k (\log k)^{1 + o(1)} \cdot l^{k - 1}\). 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.