Stronger conjecture on covering a hypergraph

A hypergraph \( H\) consists of a vertex set \( V\) together with a family \( E\) of subsets of \( V\), which are called the edges of \( H\). A \( r\)-uniform hypergraph, or \( r\)-graph, for short, is a hypergraph whose edge set consists of \( r\)-subsets of \( V\). A graph is just a special case of an \( r\)-graph with \( r=2\).

The following is a strengthened version of this conjecture.

Conjecture [1]

For every \(c>0\), there is an \(\epsilon >0\) such that if \(n\) is sufficiently large and \(\{A_i: 1 \leq i \leq cn\}\) is a collection of intersecting \(n\)-sets, then there is a set \(S\) satisfying \(|S| < n(1-\epsilon)\) and \(A_i \cap S \not = \emptyset\) for all \(1 \leq i \leq cn\).

Bibliography
1 P. Erdös, Some of my favourite unsolved problems, A Tribute to Paul Erdös, 467-478, Cambridge Univ. Press, Cambridge, 1990.