For a graph $G$ let $\tau (G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique transversal number).Is it true that if all maximal cliques in $G$ have at least $cn$ vertices then $\tau (G)=o_c(n)$?Similarly, estimate for $c>0$ the minimal $k_c(n)$ such that if every maximal clique in $G$ has at least $k_c(n)$ vertices then $\tau (G)<(1-c)n$.