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Metric embeddings traditionally study how to map n items to a target metric space such that distance lengths are not heavily distorted. However, what if we are only interested in preserving the relative order of the distances, rather than their exact lengths? In this paper, we explore the fundamental question: given triplet comparisons of the form “item i is closer to item j than to item k,” can we find low-dimensional Euclidean representations for the n items that respect those distance comparisons? Such order-preserving embeddings naturally arise in important applications—such as recommendations, ranking, crowdsourcing, psychometrics, and nearest-neighbor search—and have been studied since the 1950s under the name of ordinal or non-metric embeddings. Our main results include: Nearly-Tight Bounds on Triplet Dimension: We introduce the concept of triplet dimension of a dataset and show, surprisingly, that in order for an ordinal embedding to be triplet-preserving, its dimension needs to grow as n^2 in the worst case. This is nearly optimal, as n−1 dimensions always suffice. Tradeoffs for Dimension vs (Ordinal) Relaxation: We relax the requirement that every triplet must be exactly preserved and present almost tight lower bounds for the maximum ratio between distances whose relative order was inverted by the embedding. This ratio is known as (ordinal) relaxation in the literature and serves as a counterpart to (metric) distortion. New Bounds on Terminal and Top-k-NNs Embeddings: Moving beyond triplets, we study two well-motivated scenarios where we care about preserving specific sets of distances (not necessarily triplets). The first scenario is Terminal Ordinal Embeddings where we aim to preserve relative distance orders to k given items (the “terminals”), and for that, we present matching upper and lower bounds. The second scenario is top-k-NNs Ordinal Embeddings, where for each item we aim to preserve the relative order of its k nearest neighbors, for which we present lower bounds. To the best of our knowledge, these are some of the first tradeoffs on triplet-preserving ordinal embeddings and the first study of Terminal and Top-k-NNs Ordinal Embeddings.