Given a rectangle with dimensions $m$ inches and $n$ inches. How many points inside that rectangle have to be chosen to make sure that two of them have distance $d$ with $0 < d \leq \sqrt{2}$?
Here is my attempt: We want to divide the rectangle into "whole" unit squares each of length 1 and the remaining will be rectangles. We know we have $a = \lfloor m \rfloor \cdot \lfloor n \rfloor$ many unit squares and (EDIT:2) $b = \lceil m -\lfloor m\rfloor\rceil \cdot \lfloor n \rfloor + \lceil n -\lfloor n\rfloor\rceil \cdot \lfloor m \rfloor + \lceil m -\lfloor m\rfloor\rceil\cdot \lceil n -\lfloor n\rfloor\rceil$ many rectangles. (EDIT:1) Then choose $a + b + 1$ many points.
Feedback would be greatly appreciated.
For $m$ and $n$ integers, we need $mn+1$ points to have at least one unit square with at least two points in it
For non-integers $m$ and $n$ we need $ (\lfloor {m} \rfloor+1)(\lfloor {n}\rfloor +1)+1 $ points to cover the unit squares and the rectangles.
For $m$ integer and $n$ non-integer, we need $ (m)(\lfloor {n} \rfloor+1)+1$ Similarly for $m$ non-integer and $n$ integer we need $ (n)(\lfloor {m} \rfloor+1)+1$ points.