On maximal intersecting families of finite sets

Let r be a positive integer. A finite family $\text{H}$ of pairwise intersecting r-sets is a maximal clique of order r, if for any set A ? $\text{H}$, |A| ? r there exists a member E ? $\text{H}$ such that $\text{A ∩ E = ?}$. For instance, a finite projective plane of order r ? 1 is a maximal clique. Let N(r) denote the minimum number of sets in a maximal clique of order r. We prove $\text{N(r) ?}\text{3}\text{4}\text{r}{}^2$ whenever a projective plane of order $\text{r}\text{2}$ exists. This disproves the known conjecture N(r) ? r² ? r + 1.