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 language | English |
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Pages (from-to) | 28-38 |
Number of pages | 11 |
Journal | Journal of Graph Theory |
Volume | 49 |
Issue number | 1 |
DOIs | |
State | Published - May 2005 |
Keywords
- Chromatic number
- Independence results
- Infinite graphs
- Taylor conjecture