Ramsey numbers of trails and circuits

We show that every two‐colouring of the edges of the complete graphK n${K}_{n}$ contains a monochromatic trail or circuit of length at least2 n 2 ∕ 9 + o ( n 2 )$2{n}^{2}\unicode{x02215}9+o({n}^{2})$ , which is asymptotically best possible.