Finite subgraphs of uncountably chromatic graphs

Péter Komjáth*, Saharon Shelah

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

It is consistent that for every function f : ω → ω there is a graph with size and chromatic number N1 in which every n-chromatic subgraph contains at least f(n) vertices (n ≥ 3). This solves a $ 250 problem of Erdo″s. It is consistent that there is a graph X with Chr(X) = |X| = N1 such that if Y is a graph all whose finite subgraphs occur in X then Chr(Y) ≤ N2 (so the Taylor conjecture may fail). It is also consistent that if X is a graph with chromatic number at least N2 then for every cardinal λ there exists a graph Y with Chr(V) ≥ λ all whose finite subgraphs are induced subgraphs of X.

Original languageEnglish
Pages (from-to)28-38
Number of pages11
JournalJournal of Graph Theory
Volume49
Issue number1
DOIs
StatePublished - May 2005

Keywords

  • Chromatic number
  • Independence results
  • Infinite graphs
  • Taylor conjecture

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