Density of Monochromatic Infinite Paths

  • Allan Lo University of Birmingham
  • Nicolás Sanhueza-Matamala University of Birmingham
  • Guanghui Wang Shandong University
Keywords: Infinite paths, Ramsey theory, Density of monochromatic subgraphs


For any subset $A \subseteq \mathbb{N}$, we define its upper density to be $\limsup_{ n \rightarrow \infty } |A \cap \{ 1, \dotsc, n \}| / n$. We prove that every $2$-edge-colouring of the complete graph on $\mathbb{N}$ contains a monochromatic infinite path, whose vertex set has upper density at least $(9 + \sqrt{17})/16 \approx 0.82019$. This improves on results of Erdős and Galvin, and of DeBiasio and McKenney.

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