On the diagonal queens domination problem

It is shown that the problem of covering an n x n chessboard with a minimum number of queens on a major diagonal is related to the number-theoretic function rj(n), the smallest number of integers in a subset of {l,..., n} which must contain three terms in arithmetic progression. Several problems concerning the covering of chessboards by queens have been studied in the literature [2]. In this note we are interested in determining the minimum number of queens which can be placed on the major diagonal of an n x n chesboard and dominate (cover) all squares. Suppose that the squares are labelled (i, j), so that black and white squares have (i +j) even and odd, respectively. A subset K of N= {l,..., n} is called a diagonal dominating set if queens placed in positions {(k, k): k E K} on the black major diagonal dominate the entire board. Let diag(n) = min { IKKJ; K is a diagonal dominating set }.