Hypergraphs and various properties that we can prove about them are the basis of many techniques that are used in modern mathematics. I will mention the proof of Deducing the Density Hales-JewettHales–Jewett Theorem from an infinitary removal lemma by Tim Austin. Multidimensional SzmerediSzmerédi theorem is also be another key-word you might want to look up. The Furstenberg–Katznelson theorem (see Furstenberg-KatznelsonAn ergodic Szemerédi theorem for commuting transformations can be proven using hypergraph methods. The mathematics subject classification is 05C65.
And more importantly, take a look at Terry Tao's blog What'sWhat's new and search for "hypergraphs" to see a lot of other results that involve hypergraph methods in their proofs.
One more thing, for real world applications, hypergraph methods appear in various places including declustering problems (see declustering problemsLiu and Wu - A Hypergraph Based Approach to Declustering Problems) which are quite important to scale up the performance of parallel databases.