Multi-agent debate systems require agents to retain and revisit arguments across multiple rounds, yet limited context windows and unstructured memories make relevant information increasingly difficult to recover. Existing flat and pairwise-graph memories also struggle to capture higher-order relations, including multi-agent agreement, conflicting evidence, and evolving consensus. We introduce HEDGE, a selective memory framework that represents debate trajectories as semantic hypergraphs. Hyperedges connect related claims across agents, while intra-hyperedge relational graphs preserve fine-grained argumentative structure. A learned memory selection module (MSM), trained through supervised learning and reinforcement learning, routes each memory candidate to STORE, BACKUP, or OMIT, retaining useful information while limiting redundancy. Across seven diverse benchmarks, HEDGE consistently outperforms strong single-agent, multi-agent, and memory baselines, improving accuracy by 5.0 points on average over the strongest multi-agent system with GPT-4o-mini. The gains reach 8.2 points on conflict-heavy tasks, while HEDGE reduces memory usage by 48% and the number of debate rounds by 34%.

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