Abstract: The classical Erdős–Pósa theorem states that for each positive integer k there is an f ( k ) such that, in each graph G which does not have k + 1 disjoint cycles, there is a blocker of size at most f ( k ); that is, a set B of at most f ( k ) vertices such that G − B has no cycles. We show that, amongst all such graphs on vertex set {1,. . ., n }, all but an exponentially small proportion have a blocker of size k . We also give further properties of a random graph sampled uniformly from this class, concerning uniqueness of the blocker, connectivity, chromatic number and clique number. A key step in the proof of the main theorem is to show that there must be a blocker as in the Erdős–Pósa theorem with the extra ‘redundancy’ property that B – v is still a blocker for all but at most k vertices v ∈ B .