Ramsey numbers of trails and circuits
David Conlon, Mykhaylo Tyomkyn

TL;DR
This paper establishes a nearly optimal lower bound on the length of monochromatic trails or circuits guaranteed in any two-colouring of a complete graph, advancing understanding of Ramsey numbers for such structures.
Contribution
It provides the first asymptotically tight bound for the Ramsey numbers of trails and circuits in two-coloured complete graphs.
Findings
Monochromatic trails or circuits of length at least 2n^2/9 + o(n^2) exist in any two-colouring.
The bound is asymptotically best possible.
Advances the understanding of Ramsey numbers for paths and cycles.
Abstract
We show that every two-colouring of the edges of the complete graph contains a monochromatic trail or circuit of length at least , which is asymptotically best possible.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Computability, Logic, AI Algorithms
