Large independent sets in all large subgraphs of a random graph

Problem (proposed by Erdös and Bollobás)
In a random graph (with edge probability \( \frac{1}{2}\), find the best possible \(c \) such that every subgraph on \( n^{\alpha} \) vertices will almost surely contain an independent set of size \( c \log n \) (where \( c \) depends on \(\alpha\) ).

Bibliography
1 N. Alon, J. H. Spencer and P. Erdös, The Probabilistic Method, Wiley and Sons, New York, 1992. (Link To sample sections.)