Given $7$ distinct positive integers, prove that there is an infinite arithmetic progression of positive integers $a, a + d, a + 2d,..$ with $a < d$, that contains exactly $3$ or $4$ of the $7$ given integers.
Suppose no such d exists. Then we can split the problem into the following cases (since otherwise d=2 contradicts our assumption).
Case 1:- atleast 5 even, atmost 2 odd
Atleast 5 even no.s in the sequence are all of form 4k or all are of form 4k+2 (since otherwise d=4 contradicts our assumption). Suppose inductively atleast 5 no.s are of form 2^n+r , then atleast 5 no.s are of form 2^(n+1)+r or atleast 5 no.s are of form 2^(n+1)+2^n+r (since otherwise d=2^(n+1) contradicts our assumption). Hence these even no.s give same remainder upon division by any power of 2 which contradicts that they are distinct.
Case 2:- atleast 5 odd, atmost 2 even
The same argument as above applies here (just replace even by odd & 4k, 4k+2 by 4k+1, 4k+3).
Hence our assumption is wrong.