Vertex-disjoint sparse clique copies
= Vertex-disjoint sparse clique copies
For $p=n^{-2/3}\log n$, the expected number of pairs of $K_4$ copies sharing a vertex is
$$
O(n^7p^{12}+n^6p^{11}+n^5p^9)=o(1).
$$
Thus, with probability tending to one, all the $K_4$ copies are vertex-disjoint. Together with <sparse clique-count concentration>, this gives arbitrarily many vertex-disjoint copies with probability tending to one.