  The Hall ratio of a graph G is the maximum of |V(H)|/alpha(H) over all
subgraphs H of G. Clearly, the Hall ratio of a graph is a lower bound for the
fractional chromatic number. It has been asked whether conversely, the
fractional chromatic number is upper bounded by a function of the Hall ratio.
We answer this question in negative, by showing two results of independent
interest regarding 1-subdivisions (the 1-subdivision of a graph is obtained by
subdividing each edge exactly once).
  * For every c > 0, every graph of sufficiently large average degree contains
as a subgraph the 1-subdivision of a graph of fractional chromatic number at
least c.
  * For every d > 0, there exists a graph G of average degree at least d such
that every graph whose 1-subdivision appears as a subgraph of G has Hall ratio
at most 18.
  We also discuss the consequences of these results in the context of graph
classes with bounded expansion.
