Ramsey Numbers of Trails

Abstract

We initiate the study of Ramsey numbers of trails. Let k ≥ 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 ⌊3k/2⌋ - 1.