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We show an algorithm that finds cliques of size (log n/log log n)2 whenever a graph has a clique of size at least n/(log n)b for an arbitrary constant b. This leads to an algorithm that approximates max clique within a factor of O(n(log log n)2/(log n)3), which matches the best approximation ratio known for the chromatic number. The previously best approximation ratio known for max clique was O(n/(log n)2).
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Uriel Feige (2004) studied this question.
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