Alon–Spencer's clique-packing conjecture for random graphs

Let $G$ be the random graph $G_{n,1/2}$. Define $f(k)={n\choose k}2^{-\binom{k}{2}}$, let

$$k_0=\min\{k:f(k)<1\},$$

and set $k=k_0-4$. A packing is a collection of edge-disjoint cliques, and let $\nu_k(G)$ denote the maximum size of a packing of $k$-cliques in $G$. Since

$$\nu_k(G)\leq \frac{{n\choose 2}}{{k\choose 2}},$$

Alon–Spencer's conjecture. The trivial upper bound gives the true order of magnitude of the expected packing number:

$$\mathbb{E}\nu_k(G)=\Omega(n^2/k^2).$$

This conjecture predicts that a random graph typically admits a packing of a constant-order fraction of the maximum possible number of edge-disjoint $k$-cliques. The paper's abstract states that this conjecture is false, so the conjecture is refuted.