We introduce the notion of relative topological entropy dimension to classify the different intermediate levels of relative complexity for factor maps. By considering the dimension or "density" of special class of sequences along which the entropy is encountered, we provide equivalent definitions of relative entropy dimension. As applications, we investigate the corresponding localization theory and obtain a disjointness theorem involving relative entropy dimension.2010 Mathematics Subject Classification. Primary: 37B99, 54H20. Key words and phrases. Relative entropy dimension, relative dimension set. (*) This research is supported by NSFC(11801193). 6631 6632 XIAOMIN ZHOUfind an infinite subset W ⊂ Z + in the case of positive entropy such that along the sequence W , the symbolic names are "independent" and there exists c > 0 with lim inf n→∞ |W ∩[1,n]| n
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