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.

使用任意数量的颜色对带有顶点集\({\{1, 2, \dotsc , n\}}\)的完整图的边进行着色。最小整数f ( l ,  k ) 是多少，如果\(n > f(l,k)\)则必须存在长度为l的单调单色路径或长度为k的单调彩虹路径？ Lefmann、Rödl 和 Thomas 在 1992 年猜想\(f(l, k) = l^{k - 1}\)并证明了\(l \ge (3 k)^{2 k}\)。我们证明\(l \ge k^3 (\log k)^{1 + o(1)}\)的猜想并建立一般上限\(f(l, k) \le k (\log k )^{1 + o(1)} \cdot l^{k - 1}\)。这减少了k中从指数到多项式的最佳下限和上限之间的差距。我们还将其中一些结果推广到锦标赛设置中。