The largest set partitioned by a subfamily of a cover

AbstractDefine λ(n) to be the largest integer such that for each set A of size n and cover J of A, there exist B ⊆ A and G ⊆ J such that |B| = λ(n) and the restriction of G to B is a partition of B. It is shown that when n ⩾ 3n(1+1nn)⩽κ(n)⩽2(n − 1)(1 + 1g(n − 1) − 1g 1g(n − 1)). The lower bound is proved by a probabilistic method. A related probabilistic algorithm for finding large sets partitioned by a subfamily of a cover is presented