For a connected graph G = (V, E) of order at least two, subset T minimum total monophonic set S is forcing for if the unique containing . A cardinality S. The number ftm(S) in ftm(G) min{ftm(S)}, where taken over all sets G. We determine bounds it and find certain classes graphs. It shown that every pair a, b positive integers with 0 ≤ < ≥ a+4, there exists such mt(G) b.

It is well known to generalize the meagre ideal replacing ℵ 0 by a (regular) cardinal λ > ℵ 0 and requiring the ideal to be λ +-complete. But can we generalize the null ideal? In terms of forcing, this means finding a forcing notion similar to the random real forcing, replacing ℵ 0 by λ, so requiring it to be (< λ)-complete. Of course, we would welcome additional properties generalizing the one...