Upper bound of holographic entanglement entropy combinations

In this work, we develop a systematic formalism to evaluate the upper bound of a large family of holographic entanglement entropy combinations when fixing $n$ subsystems and fine-tuning one other subsystem. The upper bound configurations and values of these entropy combinations can be derived and classified. The upper bound of these entropy combinations reveals holographic $n+1$-partite entanglement that $n$ fixed subsystems participate in. In AdS$_3$/CFT$_2$, AdS$_4$/CFT$_3$, and even higher-dimensional holography, one can, in principle, find different formulas of upper bound values, reflecting the fundamental difference in entanglement structure in different dimensions.