#paper # Abstract We answer a question of Erd6s [1], 12] by showing that any graph of uncountable chromatic number contains an edge through which there are cycles of all (but finitely many) lengths. # Introduction Erd6s and Rado [6] gave examples of triangle free graphs of uncountable chromatic number. Erd6s and Hajnal [3] extended this by showing that, for each natural number k, there exists a graph of uncountable chromatic number containing no odd cycle of length less than k. On the other hand, Erd6s and Hajnal [4] proved that any graph of uncountable chromatic number contains all finite bipartite graphs (in particular, it contains all even cycles) and Erd6s, Hajnal and Shelah [5] showed that each such graph also contains, for some natural number k o, all odd cycles o f length greater than ko. In this note we prove the extension of this results suggested by Erd6s [l], [2] and described in the abstract. P. Erd6s (private communication) has informed the author that recently A. Hajnal also has settled this question.