Any graph of large chromatic number has an odd cycle spanning a subgraph of large chromatic number

In [1], Erdös and Hajnal raised the following question:


Is it true that for every \(k\), there is an \(f(k)\) so that if a graph \(G\) has chromatic number at least \(f(k)\), then it always contains an odd cycle whose vertices span a graph of chromatic number at least \(k\)?

1 P. Erdös, Some recent problems and results in graph theory, Discrete Math. 164 (1997) no. 1, 9396.