Is this theorem about girth and bipartite graph wrong? In this paper, the abstract mentions that the classical work of Andrásfai, Erdős, and Sós implies:

Every $n$-vertex graph with odd girth $2k+1$ and minimum degree bigger than $\dfrac{2}{2k+1}n$ must be bipartite.

I think the statement is wrong.
My idea is that if G is a graph with odd girth $2k+1$, then it contains an odd cycle (with length $2k+1$). Therefore G is not bipartite, which contradicts to the result of the statement.
Is that statement false? Or did I misunderstand something?

If you look about the middle of page 2, they define "a graph has odd girth at least $g$ if it contains no odd-length cycle of length less than $g$". 
In other words they use a concept called "odd girth" which is not the same as saying the girth is odd.
I agree it looks quite confusing though.
