On a problem about covering lines by squares

Abstract

Let S be the square [0, n]2 of side length n ∈ ℕ and let S = {S1, ..., St} be a set of unit squares lying inside S, whose sides are parallel to those of S. S is called a line cover, if every line intersecting S also intersects some Si ∈S. Let τ(n) denote the minimum cardinality of a line cover, and let τ′(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 of S. It has been conjectured by Fejes Tóth that τ(n)=2 n+O(1) and by Bárány and Füredi that τ′(n)=3/2 n+O(1). We will prove that, instead, τ′(n)=4/3 n+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 3/2 n+O(1). © 1990 Springer-Verlag New York Inc.