First, make sure that the header that you include really exists on your installation.
Second, hash_map is in the stdext namespace, make sure you make this name visible via specifying it explicitly as stdext::hash_map, or via a using directive.
Third, where are the template arguments for your hash_map in your code? I only see hash_map and not hash_map<vector<int>,string>.
Fourth, I'm not sure if std::vector can be used as a hash key for a hash map just like this. You need to define a hash function for your key (via implementing a hash_compare function object and passing it to the hash_map as third template parameter). Anyway storing these vectors by value is hardly a good idea, instead you should probably consider using pointers to these vectors as keys.