Kalai has conjectured that a simplicial complex can be partitioned
into Boolean algebras at least as roughly as a shifting-preserving
collapse sequence of its algebraically shifted complex. In
particular, then, a simplicial complex could (conjecturally) be
partitioned into Boolean intervals whose sizes are indexed by its
iterated Betti numbers, a generalization of ordinary homology Betti
numbers. This would imply a long-standing conjecture made
(separately) by Garsia and Stanley concerning partitions of
Cohen-Macaulay complexes into Boolean intervals.
We prove a relaxation of Kalai's conjecture, showing that a simplicial
complex can be partitioned into recursively defined spanning trees of
Boolean intervals indexed by its iterated Betti numbers.