The higher dimensional finiteness properties of a group generalise the notions of being finitely generated and finitely presented. Bestvina–Brady groups were introduced to settle (in the negative) the long-standing question of whether the homotopical finiteness property Fn is equivalent to its homological analogue FPn. They have since become objects of significant interest in their own right. In this talk I will outline a proof that every finitely presented Bestvina–Brady group has a quartic upper bound on its Dehn function. This bound is known to be sharp. The Dehn function of a finitely presented group measures the non-deterministic time complexity of the word problem for the group. The proof involves a new method (area-penetration pairs) for dealing with presentation with infinitely many relators.