On bounds for a board covering problem (Q1101453)

On an \(m\times n\) board, queen when placed on any square covers all the squares in the column, row and the two diagonals passing through the square. In this paper, it is shown that at least min\(\{\) m-1, [\(\frac{m+n-2}{4}]\}\) queens are necessary to cover an \(m\times n\) board, where [\(\cdot]\) is the greatest integer function. It was further shown that at most \([\frac{2}{3}n]+2\) queens cover an \(n\times n\) chessboard.5; Zbl 0325.05120); Combinatorica 1, 199-202 (1981; Zbl 0491.05044)], the vector space theorem with vector matroids [\textit{R. L. Graham}, \textit{K. Leeb}, and \textit{B. L. Rothschild}, Adv. Math. 8, 417-433 (1972; Zbl 0243.18011)]. As an example, the main theorem of the first of the cited papers by Nešetřil and Rödl may be rephrased as follows. The next two statements are equivalent for a graphical matroid A: 1) \(A=M(K_ n)\); 2) For every graphical matroid B there exists an A-Ramsey graphical matroid C.