Density of Monochromatic Infinite Subgraphs

Abstract

For any countably infinite graph G, Ramsey’s theorem guarantees an infinite monochromatic copy of G in any r-coloring of the edges of the countably infinite complete graph K. Taking this a step further, it is natural to wonder how “large” of a monochromatic copy of G we can find with respect to some measure - for instance, the density (or upper density) of the vertex set of G in the positive integers. Unlike finite Ramsey theory, where this question has been studied extensively, the analogous problem for infinite graphs has been mostly overlooked.

In one of the few results in the area, Erdős and Galvin proved that in every 2-coloring of K, there exists a monochromatic path whose vertex set has upper density at least 2/3, but it is not possible to do better than 8/9. They also showed that for some sequence εn → 0, there exists a monochromatic path P such that for infinitely many n, the set (1,2,...,n contains the first \(\left(\frac{1}{3+\sqrt{3}}-\epsilon_n\right)n\) vertices of P, but it is not possible to do better than 2n/3. We improve both results, in the former case achieving an upper density at least 3/4 and in the latter case obtaining a tight bound of 2/3. We also consider related problems for directed paths, trees (connected subgraphs), and a more general result which includes locally finite graphs for instance.

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Acknowledgements

We thank the referees for their detailed reports, especially for providing a simplification of our original proof of Theorem 1.12.

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Correspondence to Louis DeBiasio or Paul McKenney.

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Research supported in part by Simons Foundation Collaboration Grant # 283194.

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DeBiasio, L., McKenney, P. Density of Monochromatic Infinite Subgraphs. Combinatorica 39, 847–878 (2019). https://doi.org/10.1007/s00493-018-3724-2

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Mathematics Subject Classification (2010)

  • 05C55
  • 05C38