Bounded VC-Dimension Implies the Schur-Erdős Conjecture

Authors Jacob Fox, János Pach, Andrew Suk

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Jacob Fox
  • Department of Mathematics, Stanford University, Stanford, CA, USA
János Pach
  • Alfréd Rényi Institute of Mathematics, Budapest, Hungary
  • IST, Vienna, Austria
  • MIPT, Moscow, Russia
Andrew Suk
  • Department of Mathematics, University of California San Diego, La Jolla, CA, USA

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Jacob Fox, János Pach, and Andrew Suk. Bounded VC-Dimension Implies the Schur-Erdős Conjecture. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 46:1-46:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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 K_n, there is a monochromatic copy of K₃. He showed that r(3;m) ≤ O(m!), and a simple construction demonstrates that r(3;m) ≥ 2^Ω(m). An old conjecture of Erdős states that r(3;m) = 2^Θ(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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • Ramsey theory
  • VC-dimension
  • Multicolor Ramsey numbers


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