Approximation in subexponental time
One paper that gives an answer to this question is Chalermsook, Laekhanukit, &amp; Nanongkai (2013).

There are also related works in the context of Fixed Parameter Tractability such as
Hajiaghayi, Khandekar, &amp; Kortsarz (2013) and Chitnis, Hajiaghayi, Kortsarz (2013).
These hardness results are proven under various assumptions such as ETH or existence of very strong PCPs.