Collocations for equality - noun

complete, full, genuine, true | greater | economic, legal, political, racial, sexual, social

have
Women do not yet have true equality in the company.

achieve | demand, fight for, strive for | promote | establish | ensure, guarantee

equality of opportunity
People from these minority groups must have equality of opportunity.

~ between
fighting for greater equality between the sexes

~ for
the task of achieving equality for gay men

~ in
These women are demanding fairness and equality in their pay.

~ with
The women are demanding full equality with the men of their tribe.