Why doesn't greedy work for Clique?

Question

In reference to this problem (equivalently, findind a maximal clique in an undirected graph), I believe greedy approach should work (i.e. removal of people who have maximal qualms with others.). Though I cannot prove that this would always work or won't but I intuitively felt that it should, but eventually some examples could be created that fail this greedy approach.

Can someone prove/disprove whether the greedy approach is correct? Not just a counterexample, but a general explanation would be quite useful.