The cycle poset consists of the intervals of the cyclic permutation of the elements 1, 2, ..., n, ordered by inclusion. Suppose that F is a set of such intervals, none of them is a less than s others. The maximum size of F is determined under this condition. It is also shown that if the largest size of a set in this poset without containing a small subposet P is known, it solves the same problem, up to an additive constant, in the grid poset consisting of the pairs \((i,j) (1\le i,j\le n)\) and ordered coordinate-wise.
