We provide a lower bound for the coherence of the homotopy commutativity of
the Brown-Peterson spectrum, BP, at a given prime p and prove that it is at
least (2p^2 + 2p - 2)-homotopy commutative. We give a proof based on
Dyer-Lashof operations that BP cannot be a Thom spectrum associated to n-fold
loop maps to BSF for n=4 at 2 and n=2p+4 at odd primes. Other examples where we
obtain estimates for coherence are the Johnson-Wilson spectra, localized away
from the maximal ideal and unlocalized. We close with a negative result on
Morava-K-theory.Comment: This is the version published by Algebraic & Geometric Topology on 26
February 200