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Bounded VC-Dimension Implies the Schur-Erdős Conjecture

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Abstract

In 1916, Schur introduced the Ramsey number r(3; m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph Kn, there is a monochromatic copy of K3. He showed that r(3; m) ≤ O(m!), and a simple construction demonstrates that r(3; m) ≥ 2Ω(m). An old conjecture of Erdős states that r(3; m) = 2Θ(m). In this note, we prove the conjecture for m-colorings with bounded VC-dimension, that is, for m-colorings with the property that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension.

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Authors and Affiliations

  1. Stanford University, Stanford, CA, USA

    Jacob Fox

  2. Department of Mathematics, University of California at San Diego, La Jolla, CA, 92093, USA

    Andrew Suk

  3. Rényi Institute, Budapest, Hungary

    János Pach

  4. IST, Vienna, Austria

    János Pach

  5. MIPT, Moscow, Russia

    János Pach

Authors
  1. Jacob Fox
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  2. János Pach
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  3. Andrew Suk
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Corresponding author

Correspondence to Andrew Suk.

Additional information

A conference version of this paper appeared in SoCG 2020: 46, 1–8.

Supported by a Packard Fellowship and by NSF award DMS-1855635.

Partially supported by Austrian Science Fund (FWF), grant Z 342-N31.

Partially supported by the Ministry of Education and Science of the Russian Federation in the framework of MegaGrant no 075-15-2019-1926.

Partially supported by ERC Advanced Grant “GeoScape.”

Supported an NSF CAREER award, NSF grant DMS-1952786, and an Alfred Sloan Fellowship.

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Fox, J., Pach, J. & Suk, A. Bounded VC-Dimension Implies the Schur-Erdős Conjecture. Combinatorica 41, 803–813 (2021). https://doi.org/10.1007/s00493-021-4530-9

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  • DOI: https://doi.org/10.1007/s00493-021-4530-9

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