Title: Message Function Search for Hyper-relational Knowledge Graph. Abstract: Recently, the hyper-relational knowledge graph (HKG) has attracted much attention due to its widespread existence and potential applications. The pioneer works have adapted powerful graph neural networks (GNNs) to embed HKGs by proposing domain-specific message functions. These message functions for HKG embedding are utilized to learn relational representations and capture the correlation between entities and relations of HKGs. However, these works often manually design and fix structures and operators of message functions, which makes them difficult to handle complex and diverse relational patterns in various HKGs (i.e., data patterns). To overcome these shortcomings, we plan to develop a method to dynamically search suitable message functions that can adapt to patterns of the given HKG. Unfortunately, it is not trivial to design an expressive search space and an efficient search algorithm to make the search effective and efficient. In this paper, we first unify a search space of message functions that enables both structures and operators to be searchable. Especially, the classic KG/HKG models and message functions of existing GNNs can be instantiated as special cases in the proposed search space. Then, we design an efficient search algorithm to search the message function and other GNN components for any given HKGs. Through empirical study, we show that the searched message functions are data-dependent, and can achieve leading performance in link/relation prediction tasks on benchmark data sets.