A Colour Problem for Infinite Graphs and a Problem in the Theory of Relations

Bruijn, FA Dick de; Erdös, P.

doi:10.1016/s1385-7258(51)50053-7

Cited by 206 publications

(147 citation statements)

References 3 publications

Supporting

Mentioning

124

Contrasting

Unclassified

Order By: Relevance

“…de Bruijn and P. Erdős [5] implies that the chromatic number of G(n, t) is attained by a finite induced subgraph of it. So one might wonder if computing the theta function for a finite induced subgraph of G(n, t) could give a better bound than the previous corollary.…”

Section: The Theta Function Of G(n T)mentioning

confidence: 99%

Lower Bounds for Measurable Chromatic Numbers

et al. 2009

View full text Add to dashboard Cite

ABSTRACT. The Lovász theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measurable chromatic number of distance graphs on compact metric spaces.In particular we consider distance graphs on the unit sphere. There we transform the original infinite semidefinite program into an infinite, two-variable linear program which then turns out to be an extremal question about Jacobi polynomials which we solve explicitly in the limit. As an application we derive new lower bounds for the measurable chromatic number of the Euclidean space in dimensions 10, . . . , 24 and we give a new proof that it grows exponentially with the dimension.

show abstract

Section: The Theta Function Of G(n T)mentioning

confidence: 99%

Lower Bounds for Measurable Chromatic Numbers

et al. 2009

View full text Add to dashboard Cite

show abstract

“…The first is that when an infinite graph has finite homomorphic images, compactness comes to play. For example, if H is finite, then an arbitrary graph G admits a homomorphism to H if and only if all of its finite subgraphs do (see [21,51]). …”

Section: Remarksmentioning

confidence: 99%

Graph homomorphisms: structure and symmetry

1997

View full text Add to dashboard Cite

This paper is the first part of an introduction to the subject of graph homomorphism in the mixed form of a course and a survey. We give the basic definitions, examples and uses of graph homomorphisms and mention some results that consider the structure and some parameters of the graphs involved. We discuss vertex transitive graphs and Cayley graphs and their rather fundamental role in some aspects of graph homomorphisms. Graph colourings are then explored as homomorphisms, followed by a discussion of various graph products.

show abstract

“…On the one hand, for every distance graph G in R d we have χ(G) χ(R d ), where χ(G) is the usual chromatic number of the graph. On the other hand, Erdős-de Bruijn theorem (see [4]) states that χ R d = χ(H) for some finite distance graph H in R d .…”

Section: Introductionmentioning

confidence: 99%

New bounds for the distance Ramsey number

Kupavskii

Райгородский

Titova

2013

Discrete Mathematics

View full text Add to dashboard Cite

In this paper we study the distance Ramsey number R D (s, t, d). The distance Ramsey number R D (s, t, d) is the minimum number n such that for any graph G on n vertices, either G contains an induced s-vertex subgraph isomorphic to a distance graph in R d or G contains an induced t-vertex subgraph isomorphic to the distance graph in R d . We obtain the upper and lower bounds on R D (s, s, d), which are similar to the bounds for the classical Ramsey number R .

show abstract

A Colour Problem for Infinite Graphs and a Problem in the Theory of Relations

Cited by 206 publications

References 3 publications

Lower Bounds for Measurable Chromatic Numbers

Lower Bounds for Measurable Chromatic Numbers

Graph homomorphisms: structure and symmetry

New bounds for the distance Ramsey number

Contact Info

Product

Resources

About