On a problem about covering lines by squares (Q6377050)

Let S be a square of side length \(n\in {\mathbb{N}}\), and let \({\mathcal S}=\{S_ 1,...,S_ t\}\) be a collection of unit squares contained in S and with sides parallel to those of S. \({\mathcal S}\) is called a line cover if every line intersecting S also intersects some \(S_ i\in {\mathcal S}\). Let \(\tau '(n)\) denote the minimum of t such that there is a line cover \({\mathcal S}\) of cardinality t. It is proved that \(\tau '(n)=4n/3+O(1).\) Some related questions are discussed.