So, given any set of three integers, prove there is a pair whose sum us even, and then prove or disprove that there is a pair whose sum is odd.
To prove that there is a pair whose sum is even, couldn't I say that since there are 3 integers that are either even or odd, there must be 2 that are even, or 2 that are odd, in which the sum of the even pair or odd pair is even?
For the second part, I know that there can be a few possibilities for an odd sum, but that is dependent upon the set of integers, so I'm not sure how to exactly prove that.
According to pigeonhole principle, since there are $3$ integers (pigeons) and each of which can be even or odd ($2$ holes), either there are $\lceil \dfrac{3}{2}\rceil = 2$ odd or $2$ even integers, both of which give an even sum.
However, there is a possibility that all are even, therefore a pair of integers with odd sum can't be guaranteed.