Let $\Sigma$ be an alphabet of size $|\Sigma|=k$. Let $w\in\Sigma^*$ be a word over $\Sigma$. If $|w| > 2^k$, then $w$ contains an infix $y$ with $|y|\ge 2$, such that every letter occurring in y occurs an even number of times in $y$.
Seems it's obvious but I can't prove it.
On
Maybe this should be a comment to Henning's answer, but I think there is still a crucial point of the reasoning to be explained, namely...
For each prefix of x associate a k-tuple over {0,1} that represents the parity of each of the k letters in that prefix. Obviously, there are only $2^k$ such binary k-tuples possible, so if there are $2^k+1$ letters in x, one of the k-tuples must repeat at two different prefixes -- there's the pigeonhole principle. The points at which they repeat give Henning's two prefixes in which the parity of each letter is the same.
By the pigeonhole principle, there must be two different prefixes of $w$ that contain the same number of each letter modulo 2. Their difference is your $y$.