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Distance Preserving Ramsey Graphs

Published online by Cambridge University Press:  23 April 2012

DOMINGOS DELLAMONICA Jr  and
VOJTĚCH RÖDL
DOMINGOS DELLAMONICA Jr
Affiliation:
Department of Mathematics and Computer Science, Emory University, 400 Dowman Dr., W401, Atlanta, GA 30322, USA (e-mail: ddellam@mathcs.emory.edu, rodl@mathcs.emory.edu)
VOJTĚCH RÖDL
Affiliation:
Department of Mathematics and Computer Science, Emory University, 400 Dowman Dr., W401, Atlanta, GA 30322, USA (e-mail: ddellam@mathcs.emory.edu, rodl@mathcs.emory.edu)
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Abstract

We prove the following metric Ramsey theorem. For any connected graph G endowed with a linear order on its vertex set, there exists a graph R such that in every colouring of the t-sets of vertices of R it is possible to find a copy G* of G inside R satisfying:

  • • distG*(x, y) = distR(x, y) for every x, y ∈ V(G*);

  • • the colour of each t-set in G* depends only on the graph-distance metric induced in G by the ordered t-set.

Keywords

Primary 05C55
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 4 , July 2012 , pp. 554 - 581
Copyright
Copyright © Cambridge University Press 2012

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