Given $7$ distinct positive integers, prove that there is an infinite arithmetic progression (AP) of positive integers $a, a + d, a + 2d,..$ with $a < d$, that contains exactly $3$ or $4$ of the $7$ given integers.
I could not think of anything much, I just rephrased the problem as :- we have to find a $d$ such that out of the given $7$ numbers, $3$ or $4$ are same in modulo $d$.
Divide the given integers into two subsets; evens and odds. Then one of the two contains at least four integers. If one contains precisely four integers, take $(a,d)=(0,2)$. If one contains more than four integers, divide it into two subsets by its remainders mod $4$, and so forth.