On a problem about covering lines by squares

Abstract

LetS be the square [0,n]² of side lengthn ∈ ℕ and letS = {S ₁, ...,S _t} be a set of unit squares lying insideS, whose sides are parallel to those ofS.S is called a line cover, if every line intersectingS also intersects someS _i ∈S. Letτ(n) denote the minimum cardinality of a line cover, and letτ′(n) be defined in the same way, except that we restrict our attention to lines which are parallel to either one of the axes or one of the diagonals ofS. It has been conjectured by Fejes Tóth thatτ(n)=2n+O(1) and by Bárány and Füredi thatτ′(n)=3/2n+O(1). We will prove that, instead,τ′(n)=4/3n+O(1) and, as to Fejes Tóth's conjecture, we will exhibit a “noninteger” solution to a related LP-relaxation, which has size equal to 3/2n+O(1).