Density of Monochromatic Infinite Paths

Allan Lo, Nicolás Sanhueza-Matamala, Guanghui Wang

Abstract


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.


Keywords


Infinite paths; Ramsey theory; Density of monochromatic subgraphs

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