Recently, I was going over a curious recreational math paper titled "On the Diagonal Queens Domination Problem". The main result of the paper is establishing the minimum value $diag(n)$ of number of queens needed to be kept on the diagonal of a $n \times n$ chessboard so that they attack all the squares on the board.
The authors make use of some clever lemma and make an observation that $diag(n) = n + 1 - r_3(\lceil \frac{n}{2} \rceil)$. Here, $r_3(.)$ denotes the minimum value $k$ such that any $k$-element subset of $[n]$ contains a $k$ term arithmetic progression (a la Roth's theorem statement). In order to prove this statement, authors make use of a number theoretic statement which they do not prove. I also cannot prove i by myself because of my limited number theory background (and maybe because I am dense). The statement says the following
A collection of $k$ even numbers or $k$ odd numbers (the numbers in the collection have same parity, this will keep their sum even) is midpoint free even sum subset of $[n]$ $\mathit{if\ and\ only\ if}$ all the half of the numbers in the collection (rounded down if not integer) is a midpoint-free subset of $\lceil \frac{n}{2} \rceil$.
A few definitions to clarify the question. A subset $X$ of $[n]$ is midpoint free if for any two elements $i,j \in X$ $(i+j)/2 \not\in X$. A subset is even sum if $i+j$ is even for any $2$ elements $i,j \in X$.
I hope the question is clear. In case, its not, I apologize and would request you to kindly let me know whats not clear.