solutions of a particular equation

A-ManESL

Hello all... I have a problem which I have been grappling with for some time. Let b be a positive integer and consider the equation z = x + y + b where x,y,z are variables. Suppose the integers {1,2,...4b+5} are partitioned in two classes. I wish to show that at least one of the classes contains a solution to the equation.

I have tried using induction on b. The case b = 1 has been solved entirely by me. But I cannot understand how to use the induction hypothesis to prove the result. The more I think of it, the more I feel that a different approach to the problem is needed, but I cant figure out what. It is sort of a special case of a research problem, which has been solved in a more general way. I have little experience of doing research on my own, and so will be glad if anyone can offer me any advice or hints. Thanks.

disregardthat

A class C of {1,2,...,4b+5} contains a solution is equivalent to that x,y,z are elements of C ?

ramsey2879

If you give us your solution for the case b = 1, maybe someone could generalize it for the other cases. Oddly with one less or one more variable, i.e. z = w+x + y +1 or z = x+1, there is a simple counterexample.