Monochromatic paths in random tournaments

Abstract

We prove that, with high probability, any 2‐edge‐coloring of a random tournament on n vertices contains a monochromatic path of length Ω(n/log⁡n). This resolves a conjecture of Ben‐Eliezer, Krivelevich, and Sudakov and implies a nearly tight upper bound on the oriented size Ramsey number of a directed path.

On-line size Ramsey number for monotone k-uniform ordered paths with uniform looseness

“…Their lower bound for S ori is higher than their upper bound for S dir (Bucic, Letzter, and Sudakov [7] improved their upper bound on S ori ). where C 2 is an absolute constant, but C 1 (t) depends on t. They require C 1 (t) < C 1/(t−1) 8(2t − 2) t−1 (16(t − 1) 2 ) t for some absolute constant C. Therefore, their lower bound is at most 1 (2t) 3t (m + 1) 2t−2 (log(m + 1))…”

Directed Ramsey number for trees

“…In particular, the answer to how large a monochromatic tree we can find in an edge colouring of a tournament T on n vertices, depends on T as well as the colouring. For example, if T is the transitive tournament on n vertices, there is a 2-edge-colouring with no monochromatic directed path of length √ n. Contrasting this, in a recent paper [6], we prove that if T is a random tournament on n vertices then, with high probability, in every 2-edge-colouring of T there is a monochromatic path of length at least cn √ log n , where c > 0 is an absolute constant. Keeping this in mind, an underlying structure more analogous to undirected Ramsey case is the complete directed graph on n vertices, which we denote by ← → K n .…”

Directed Ramsey number for trees

“…In particular, the answer to how large a monochromatic tree we can find in an edge colouring of a tournament T on n vertices, depends on T as well as the colouring. For example, if T is the transitive tournament on n vertices, there is a 2-colouring with no monochromatic directed path of length √ n. Contrasting this, in a recent paper [6], we prove that if T is a random tournament on n vertices then, with high probability, in every 2-colouring of T there is a monochromatic path of length at least cn Very few directed Ramsey numbers are known; here we outline some of the few results in this area. Harary and Hell [19] introduced the notion of directed Ramsey numbers (for two colours) and determined its value for certain small graphs.…”