Developing an idea of M. Gromov (1993), we study the intersection formula for random subsets with density. The density of a subset A in a finite set E is defined by dens A WD log jEj .jAj/. The aim of this article is to give a precise meaning of Gromov's intersection formula: "Random subsets" A and B of a finite set E satisfy dens.A \ B/ D dens A C dens B 1.As an application, we exhibit a phase transition phenomenon for random presentations of groups at density =2 for any 0 < < 1, characterizing the C 0 . /-small cancellation condition. We also improve an important result of random groups by G. Arzhantseva and A. Ol'shanskii (1996) from density 0 to density 0 Ä d < 1=.120m 2 ln.2m//.

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