Which of the following rules is key to preventing slips and falls around the pool?

Preventing slips and falls around the pool is crucial for ensuring the safety of all patrons. The rule that emphasizes "no diving in shallow areas" directly addresses the risk associated with diving into water that isn’t deep enough. This can lead to serious injuries, including head and neck injuries, which can occur if someone dives into shallow water and hits the bottom. By enforcing this rule, the risk of accidents is significantly reduced, as it discourages behavior that could lead to dangerous situations at the pool's edge.

The other options do not promote safety around the pool area. Encouraging running can lead to slips and falls, which directly contradicts the goal of maintaining a safe environment. Allowing horseplay also increases the risk of accidents, as it often leads to unpredictable movements that can result in falls or collisions. Prohibiting swimming would not inherently prevent slips and falls, nor would it address the behaviors that contribute to poolside accidents. Therefore, emphasizing the rule against diving in shallow areas is essential for maintaining safety and preventing injuries around the pool.