Using the Ramsey Theorem
Let $X$ be some countably infinite set and colour the elements of $X(n)$ (the subsets of $X$ of size $n$) in $c$ different colours. Then there exists some infinite subset $M\subset X$ such that the size $n$ subsets of $M$ all have the same colour.
it's easy to prove the following:
Given a countably infinite set of infinite binary strings $X$, and a natural number $n$, then there exists a infinite subset $Y\subset X$ such that no subset of $Y$ with $n$ elements has XOR sum zero
My question is: there's a way to prove this without using Ramsey?