On the diagonal queens domination problem (Q1073797)

Let the squares of an \(n\times n\) chessboard be labelled (i,j), so that black and white squares have \(i+j\) even and odd, respectively. A subset K of \(N=\{1,...,n\}\) is a diagonal dominating set if queens placed in positions \(\{\) (k,k): \(k\in K\}\) dominate the entire board. A subset X of N is called midpoint-free if for all (i,j)\(\subseteq X\), \((i+j)/2\not\in X\), and X is an even-sum subset if its elements are either all odd or all even. It is shown that K is a diagonal dominating set if and only if N-K is a midpoint-free, even-sum set. If diag (n)\(=\min \{| K|:\) K is a diagonal dominating set\(\}\) and \(r_ 3(n)\) is the smallest k such that any k-subset of N contains a 3-term arithmetic progression, then diag (n)\(=n+1-r_ 3([n/2])\).