Graphs without short odd cycles are nearly bipartite

It is proved that for every constant ϵ > 0 and every graph G on n vertices which contains no odd cycles of length smaller than ϵn, G can be made bipartite by removing (15/ϵ)ln(10/ϵ)) vertices. This result is best possible except for a constant factor. Moreover, it is shown that one candestroy all odd cycles in such a graph G also by omitting not more than (200/ϵ2)(ln(10/ϵ))2 edges.