Negative partition relations for ordinals omega(omega alpha)

For ordinals alpha, beta, and gamma, the expression alpha negated right arr ow(beta, gamma)(2) means there is a partition of the pairs from alpha, [alp ha](2) = Delta(0) boolean OR Delta(1) such that for any X subset of or equa l to alpha, if the order type of X is beta then [X](2) not subset of or equ al to Delta(0) and if the order type of X is gamma then [X](2) not subset o f or equal to Delta(1). It is shown that if alpha < omega(1), is multiplica tively decomposable, then omega(alpha) negated right arrow (omega(alpha), n )(2) for n =4 or n = 6, depending on the degree of decomposability of alpha . (C) 1999 Academic Press.