Unboundedness for Generalized Odd Cyclic Transversality

All graphs considered in this note are finite and without multiple edges and, unless the contrary is clear from the context, without loops. In 1962 (1984) P. Erdos and L. P6sa (V. Neumann-Lara) proved the following Conjecture(s) [1,3] ([4]) : For every positive integer k there exists another positive integer k' such that for every graph G, either (i) G has k vertex-disjoint (long and/or even) cycles or (ii) there exists a subset X of less than k' vertices of G such that G \ X has no (long and/or even) cycles. By "long" cycles we mean cycles of length :2: p" where p, is a fixed positive integer. In fact, Neumann-Lara proved his cases by means of Menger and Ramsey Theorems. We assert that, when replacing in this Erd5s-P6sa Theorem the term "cycle", then the modified conjecture becomes false. In fact, we have the following.