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In knowledge graph completion (KGC), predicting triples involving emerging entities and/or relations, which are unseen when the KG embeddings are learned, has become a critical challenge. Subgraph reasoning with message passing is a promising and popular solution. Some recent methods have achieved good performance, but they (i) usually can only predict triples involving unseen entities alone, failing to address more realistic fully inductive situations with both unseen entities and unseen relations, and (ii) often conduct message passing over the entities with the relation patterns not fully utilized. In this study, we propose a new method named RMPI which uses a novel Relational Message Passing network for fully Inductive KGC. It passes messages directly between relations to make full use of the relation patterns for subgraph reasoning with new techniques on graph transformation, graph pruning, relation-aware neighborhood attention, addressing empty subgraphs, etc., and can utilize the relation semantics defined in the ontological schema of KG. Extensive evaluation on multiple benchmarks has shown the effectiveness of techniques involved in RMPI and its better performance compared with the existing methods that support fully inductive KGC. RMPI is also comparable to the state-of-the-art partially inductive KGC methods with very promising results achieved. Our codes and data are available at https://github.com/zjukg/RMPI.