We consider the problem of clustering with user feedback. Existing methods express constraints about the input data points, most commonly through must-link and cannot-link constraints on data point pairs. In this paper, we introduce existential cluster constraints: a new form of feedback where users indicate the features of desired clusters. Specifically, users make statements about the existence of a cluster having (and not having) particular features. Our approach has multiple advantages: (1) constraints on clusters can express user intent more efficiently than point pairs; (2) in cases where the users' mental model is of the desired clusters, it is more natural for users to express cluster-wise preferences; (3) it functions even when privacy restrictions prohibit users from seeing raw data. In addition to introducing existential cluster constraints, we provide an inference algorithm for incorporating our constraints into the output clustering. Finally, we demonstrate empirically that our proposed framework facilitates more accurate clustering with dramatically fewer user feedback inputs.