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Erd\H{o}s-Ko-Rado (EKR) type theorems yield upper bounds on the sizes offamilies of sets, subject to various intersection requirements on the sets inthe family. Stability versions of such theorems assert that if the size of afamily is close to the maximum possible size, then the family itself must beclose (in some appropriate sense) to a maximum-sized family.  In this paper, we present an approach to obtaining stability versions ofEKR-type theorems, via isoperimetric inequalities for subsets of the hypercube.Our approach is rather general, and allows the leveraging of a wide variety ofexact EKR-type results into strong stability versions of these results, withoutgoing into the proofs of the original results.  We use this approach to obtain tight stability versions of the EKR theoremitself and of the Ahlswede-Khachatrian theorem on $t$-intersecting families of$k$-element subsets of $\{1,2,\ldots.n\}$ (for $k < \frac{n}{t+1}$), and toshow that, somewhat surprisingly, all these results hold when the intersectionrequirement is replaced by a much weaker requirement.  Other examples include stability versions of Frankl's recent result on theErd\H{o}s matching conjecture, the Ellis-Filmus-Friedgut proof of theSimonovits-S\'{o}s conjecture, and various EKR-type results on $r$-wise(cross)-$t$-intersecting families.

Stability versions of Erd\H{o}s-Ko-Rado type theorems, via isoperimetry