A slightly simpler proof of Kostochka's lemma. It avoids the need for one extra lemma.

For a collection of maximum cliques Q in a graph G, let X_Q be the
intersection graph of Q.

Lemma. If Q is a collection of maximum cliques in a graph G with
\omega(G) > 2/3 (\Delta(G) + 1) such that X_Q is connected, then \cap
Q \neq \emptyset.

Proof. Suppose not and choose a counterexample Q := {Q_1, ..., Q_r}
minimizing r.

Let A be a noncutvertex in X_Q and B a neighbor of A. Put Z := Q -
{A}. Then X_Z is connected and hence by minimality of r, \cap Z \neq
\emptyset. In particular, |\cup Z| \leq \Delta(G) + 1.

Hence |\cup Q| \leq |\cup Z| + |A - B| \leq 2(\Delta(G) + 1) -
\omega(G) < 2\omega(G). This contradicts Hajnal's lemma.