For subadditive valuations, an allocation always exists in which each of n agents receives at least 1/O((log log n)^2) of her maximin share, improving the previous 1/O(log n log log n) bound.
A constant-factor approximation for nash social welfare with subadditive valuations
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Beating the Logarithmic Barrier for the Subadditive Maximin Share Problem
For subadditive valuations, an allocation always exists in which each of n agents receives at least 1/O((log log n)^2) of her maximin share, improving the previous 1/O(log n log log n) bound.