Maximal Chains and Antichains in Boolean Lattices

The following equivalent results in the Boolean lattice 2 are proven. (a) Every fibre of 2 contains a maximal chain. (b) Every cutset of 2 contains a maximal antichain. (c) Every red-blue colouring of the vertices of 2 produces either a red maximal chain or a blue maximal antichain. (d) Given any n antichains in 2 there is a disjoint maximal antichain. Statement (a) is then improved to: (a') Every fibre of 2 contains at least n!/2"-maximal chains. One conjecture of Lonc and Rival is supported, and another conjecture disproved, by showing: (i) Every fibre of 2" has order r(1.25 n) elements. (ii) There is a minimal fibre of 2 "(n >_-4) of size 2 + 2.