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We consider a finite simplicial complex   $K$   together with its successive barycentric subdivisions   ${\text{Sd}}^d(K), d\geq0,$   and study the expected topology of a random subcomplex in   ${\text{Sd}}^d(K), d\gg0$  . We get asymptotic upper and lower bounds for the expected Betti numbers of those subcomplexes, together with the average Morse inequalities and expected Euler characteristic.

Asymptotic Topology of Random Subcomplexes in a Finite Simplicial Complex