  We develop a scheme for generating a universal qubit cluster state using
probabilistic Bell measurements without the need for feed-forward or long-time
quantum memories. Borrowing ideas from percolation theory we numerically show
that using unambiguous Bell measurements that succeed with 75% success
probability, one could build a cluster state with an underlying pyrochlore
geometry such that the probability of having a spanning cluster in a chosen
direction approaches unity in the limit of an infinite lattice size. The
initial resources required for the generation of a universal state in our
protocol are 3-qubit cluster states that are within experimental reach and are
a minimal resource for a Bell-measurement-based percolation proposal. Since
single and multi-photon losses can be detected in Bell measurements, our
protocol raises the prospect of a fully error-robust scheme.
