Unbeatable clique covers

Let $G=(V,E)$ be a simple, undirected graph. A clique cover is a set ${\cal C}\subseteq {\cal P}(V)$ such that

1. every element of ${\cal C}$ is a clique, and
2. $\bigcup {\cal C} = V$.

We call a clique cover ${\cal C}$ beatable if there is a clique cover ${\cal C}_1$ such that $$|{\cal C}_1 - {\cal C}| < |{\cal C} - {\cal C}_1|.$$ Non-beatable clique covers are called unbeatable.

It is easy to prove that every finite graph has an unbeatable clique cover. But what about infinite graphs?

• what is $\mathcal{C-C_1}$ and $\mathcal{C_1-C}$ here? – user94040 Dec 14 '16 at 8:59
• @AJ. He means the set difference. So, the condition states that in going from $\cal C$ to $\cal C_1$, you've removed more cliques than you've added. – Joel David Hamkins Dec 14 '16 at 13:30