eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-06-08
46:1
46:8
10.4230/LIPIcs.SoCG.2020.46
article
Bounded VC-Dimension Implies the Schur-Erdős Conjecture
Fox, Jacob
1
Pach, János
2
3
4
Suk, Andrew
5
Department of Mathematics, Stanford University, Stanford, CA, USA
Alfréd Rényi Institute of Mathematics, Budapest, Hungary
IST, Vienna, Austria
MIPT, Moscow, Russia
Department of Mathematics, University of California San Diego, La Jolla, CA, USA
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol164-socg2020/LIPIcs.SoCG.2020.46/LIPIcs.SoCG.2020.46.pdf
Ramsey theory
VC-dimension
Multicolor Ramsey numbers