The (t-1)-chromatic Ramsey number for paths

Bucić, Matija; Khamseh, Amir

doi:10.48550/arxiv.2101.00779

Search citation statements

Order By: Relevance

Paper Sections

Select...

Introduction1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2022

Publication Types

Select...

Other1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

(1 citation statement)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The set-coloring Ramsey number and its special cases have been studied by many researchers for over half a century (see [1,2,3,7,10,14,22,23]). Indeed, note that when s = 1, R(n; r, 1) = R(n; r) is just the classical Ramsey number.…”

Section: Introductionmentioning

confidence: 99%

Set-coloring Ramsey numbers via codes

Conlon¹,

Fox²,

He³

et al. 2022

Preprint

View full text Add to dashboard Cite

For positive integers n, r, s with r > s, the set-coloring Ramsey number R(n; r, s) is the minimum N such that if every edge of the complete graph KN receives a set of s colors from a palette of r colors, then there is guaranteed to be a monochromatic clique on n vertices, that is, a subset of n vertices where all of the edges between them receive a common color. In particular, the case s = 1 corresponds to the classical multicolor Ramsey number. We prove general upper and lower bounds on R(n; r, s) which imply that R(n; r, s) = 2 Θ(nr) if s/r is bounded away from 0 and 1. The upper bound extends an old result of Erdős and Szemerédi, who treated the case s = r − 1, while the lower bound exploits a connection to error-correcting codes. We also study the analogous problem for hypergraphs.

show abstract