Clique transversal number is sublinear for graphs with no small cliques

Let \( \tau(G)\) denote the cardinality of a smallest set of vertices in \( G\) that shares some vertex with every clique of \( G\).

Problem [1]

Suppose all cliques in \(G\) have \(cn\) vertices. Is it true that \[ \tau(G) = o(n)? \]

1 P. Erdös. Problems and results on set systems and hypergraphs, Extremal problems for finite sets (Visegrád, 1991), Bolyai Soc. Math. Stud., 3, pp. 217-227, János Bolyai Math. Soc., Budapest, 1994.