TITLE: Erdős's exercise.
QUESTION [8 upvotes]: I have tried to solve an exercise I saw in "Topics in the theory of numbers" (Erdős & Suranyi)
many times but failed every time I tried.
Here it is: 
Prove that if $a_1,a_2,\cdots$ is an infinite sequence whose elements are either $1$ or $-1$, then for every positive $K$, there exist numbers $b,c,d$ for which $$\left |\sum\limits_{i=k}^d a_{ib+c}\right|>K$$
But I read today that a very similar problem is open. Look at this abstract.
So, Erdős  suggested as an exercise an open problem?
Or if not can anybody give me a proof of the above statement?
Thanks in advance.

REPLY [3 votes]: It is exercise 12 of chapter 6.19, where the problem is given as a "simple and interesting application of [Van der Waerden's theorem on arithmetic progressions]".
http://books.google.com/books?id=MTefj5-5OwEC&pg=PA198
There is a typo in the book's statement of the exercise that makes it ill-formed.  Probably the idea is to use Van der Waerden's theorem to prove that discrepancy of arithmetic progressions in $\lbrace 1,\dots, n \rbrace$ is unbounded as a function of $n$.   This is, as they say, a simple application because for large $n$ either the set of $+1$'s or the set of $-1$'s contains (by VdW) an arithmetic progression of length greater than $K$.