Steiner determines the maximum ratio between fractional chromatic number and Hall ratio as (log n)^(1-o(1)), and constructs graphs with bounded Hall ratio, arbitrarily large fractional chromatic number, and every subgraph containing an independent set touching a constant fraction of its edges.
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Fractional chromatic number vs. Hall ratio
Steiner determines the maximum ratio between fractional chromatic number and Hall ratio as (log n)^(1-o(1)), and constructs graphs with bounded Hall ratio, arbitrarily large fractional chromatic number, and every subgraph containing an independent set touching a constant fraction of its edges.