Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 61: Issue 1
Set-Coloring Ramsey Numbers via Codes
Authors:
David Conlon Department of Mathematics, California Institute of Technology, Pasadena, CA 91125, USA

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Jacob Fox Department of Mathematics, Stanford University, Stanford, CA 94305, USA

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Xiaoyu He Department of Mathematics, Princeton University, Princeton, NJ 08544, USA

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Dhruv Mubayi Department of Mathematics, Statistics and Computer Science, University of Illinois, Chicago, IL 60607, USA

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Andrew Suk Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, USA

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Jacques Verstraรซte Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, USA

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Pages:
1โ€“15
Online Publication Date:
28 Mar 2024
Publication Date:
15 Apr 2024
Article Category:
Research Article
DOI:
https://doi.org/10.1556/012.2023.04302
Keywords:
Ramsey numbers; set colorings; error-correcting codes; stepping-up lemma
Restricted access

For positive integers ๐‘›, ๐‘Ÿ, ๐‘  with ๐‘Ÿ > ๐‘ , the set-coloring Ramsey number ๐‘…(๐‘›; ๐‘Ÿ, ๐‘ ) is the minimum ๐‘ such that if every edge of the complete graph ๐พ๐‘ receives a set of ๐‘  colors from a palette of ๐‘Ÿ colors, then there is guaranteed to be a monochromatic clique on ๐‘› vertices, that is, a subset of ๐‘› vertices where all of the edges between them receive a common color. In particular, the case ๐‘  = 1 corresponds to the classical multicolor Ramsey number. We prove general upper and lower bounds on ๐‘…(๐‘›; ๐‘Ÿ, ๐‘ ) which imply that ๐‘…(๐‘›; ๐‘Ÿ, ๐‘ ) = 2ฮ˜(๐‘›๐‘Ÿ) if ๐‘ /๐‘Ÿ is bounded away from 0 and 1. The upper bound extends an old result of Erdล‘s and Szemerรฉdi, who treated the case ๐‘  = ๐‘Ÿ โˆ’ 1, while the lower bound exploits a connection to error-correcting codes. We also study the analogous problem for hypergraphs.

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Cover Studia Scientiarum Mathematicarum Hungarica
Studia Scientiarum Mathematicarum Hungarica
Print ISSN:
0081-6906
Online ISSN:
1588-2896

Editors in Chief

Gábor SIMONYI (Rényi Institute of Mathematics)
András STIPSICZ (Rényi Institute of Mathematics)
Géza TÓTH (Rényi Institute of Mathematics) 

Managing Editor

Gábor SÁGI (Rényi Institute of Mathematics)

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  • Imre BÁRÁNY (Rényi Institute of Mathematics)
  • Károly BÖRÖCZKY (Rényi Institute of Mathematics and Central European University)
  • Péter CSIKVÁRI (ELTE, Budapest) 
  • Joshua GREENE (Boston College)
  • Penny HAXELL (University of Waterloo)
  • Andreas HOLMSEN (Korea Advanced Institute of Science and Technology)
  • Ron HOLZMAN (Technion, Haifa)
  • Satoru IWATA (University of Tokyo)
  • Tibor JORDÁN (ELTE, Budapest)
  • Roy MESHULAM (Technion, Haifa)
  • Frédéric MEUNIER (École des Ponts ParisTech)
  • Márton NASZÓDI (ELTE, Budapest)
  • Eran NEVO (Hebrew University of Jerusalem)
  • János PACH (Rényi Institute of Mathematics)
  • Péter Pál PACH (BME, Budapest)
  • Andrew SUK (University of California, San Diego)
  • Zoltán SZABÓ (Princeton University)
  • Martin TANCER (Charles University, Prague)
  • Gábor TARDOS (Rényi Institute of Mathematics)
  • Paul WOLLAN (University of Rome "La Sapienza")

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Studia Scientiarum Mathematicarum Hungarica
Language English
French
German
Size B5
Year of
Foundation		 1966
Volumes
per Year		 1
Issues
per Year		 4
Founder Magyar Tudományos Akadémia  
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
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ISSN 0081-6906 (Print)
ISSN 1588-2896 (Online)

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