Properly colored Hamilton cycles in edge-colored complete graphs

It is shown that, for ϵ>0 and n>n0(ϵ), any complete graph K on n vertices whose edges are colored so that no vertex is incident with more than (1-1/√2-ε)n edges of the same color contains a Hamilton cycle in which adjacent edges have distinct colors. Moreover, for every k between 3 and n any such K contains a cycle of length k in which adjacent edges have distinct colors.