On the Turán number of the directed path

Daniel P. Johnston; Cory Palmer; Amites Sarkar

On the Turán number of the directed path

Combinatorics 2026-07-10 v1

Authors: Daniel P. Johnston , Cory Palmer , Amites Sarkar

Abstract

In this note we determine the maximum number of arcs in a digraph on $n$ vertices that does not contain a length- $k$ directed path $\overrightarrow{P}_{k+1}$ for $n \geq 50k^6$ . This improves a theorem of Zhou and Li [ $\textit{Graphs Combin.}$ 39(3), 2023] who proved the result with a superexponential threshold on $n$ .

Keywords

Turán number directed path digraph extremal graph theory

Cite

@article{arxiv.2607.10036,
  title  = {On the Turán number of the directed path},
  author = {Daniel P. Johnston and Cory Palmer and Amites Sarkar},
  journal= {arXiv preprint arXiv:2607.10036},
  year   = {2026}
}