Erdős's exercise. 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.
 A: This follows from a theorem of Roth. You can read his proof at [1].
He shows not only that it grows unbounded, but we can get a lower bound on the order of growth. Roughly speaking, we can guarantee discrepancy on the order of $O(n^{1/4})$. 
The Erdos Discrepancy problem, which is the open conjecture mentioned in a paper you linked to, is the 5th polymath project, so some information is available there. In particular, there is an explicit solution to the problem you ask written by Tim Gowers.

References
[1]: Roth, K. F., Remark concerning integer sequences. Acta Arith. 9 1964 257--260. MR0168545 (29 #5806) 
A: 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 $K$.
