Ramsey Numbers of Trails

We initiate the study of Ramsey numbers of trails. Let $k \geq 2$ be a positive integer. The Ramsey number of trails with $k$ vertices is defined as the the smallest number $n$ such that for every graph $H$ with $n$ vertices, $H$ or the complete $\overline{H}$ contains a trail with $k$ vertices. We prove that the Ramsey number of trails with $k$ vertices is at most $k$ and at least $2\sqrt{k}+\Theta(1)$. This improves the trivial upper bound of $\lfloor 3k/2\rfloor -1$.