Estimate the clique transversal number of a graph

A problem on clique transversals (proposed by Erdös, Gallai and Tuza[1])

Estimate the cardinality, denoted by \(\tau(G)\), of a smallest set of vertices in \(G\) that shares some vertex with every clique of \(G\).

Denote by \( R(n)\) the largest integer such that every triangle-free graph on \( n\) vertices contains an independent set of \( R(n)\) vertices. Is it true that \( \tau(G) \leq n -R(n)\)?

From the results on the Ramsey numbers \( r(3,k)\), we know that \( c \sqrt {n \log n} < R(n) < c' \sqrt {n \log n}\). So far, the best current bound [1] is \( \tau(G) \leq n- \sqrt{2n}+c\) for a small constant \( c\).

1 P. Erdös, T. Gallai and Zs. Tuza, Covering the cliques of a graph with vertices,Topological, Algebraical and Combinatorial Structures, Frolík's memorial volume, Discrete Math. 108 (1992), 279-289.