Dialogue state tracking is an essential component in multi-domain dialogue systems that aims to accurately determine the current dialogue state based on the dialogue history. Existing research has addressed the issue of multiple mappings in dialogues by employing slot self-attention as a data-driven approach. However, learning the relationships between slots from a single sample often has limitations and may introduce noise and have high time complexity issues. In this paper, we propose an external slot relation memory-based dialogue state tracking model (ER-DST). By utilizing external memory storage, we learn the relationships between slots as a dictionary of multi-domain slot relations. Additionally, we employ a small filter to discard slot information irrelevant to the current dialogue state. Our method is evaluated on MultiWOZ 2.0 and MultiWOZ 2.1, achieving improvements of 0.23% and 0.39% over the baseline models, respectively, while reducing the complexity of the slot relationship learning component to O(n).