By Neil H. Williams

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In particular then, K ( K , K ) ' . + + We conclude this list of negative relations with the following theorem. 9 (GCH). Z ~ K is singular then K + P (K+, (3)K')'. hoof. By the GCH, I [ K + ] ~ I = K + so we may put [ K + ] ~= { A , ; a < K + } . For e a c h a , p u t d , = { A p ; P < a a n d A p C a } , s o ld,I<~and [ K + ] ~ = u { d a ; a < K ' } . Choose an increasing sequence of cardinals ( K , ; u < K ' ) all less than K such that K = Z(K,,; u < K ' ) . Choose subfamilies d,,of d ,with I d a u I S ~ , , such that d, = U{ d a uu;< K ' } , for each a.

6 51 Define a partition { r k ; k < y} of ["213 as follows: rk = A: if 2 < k < 7 , rl = K U ~A T ,~ ro = ["213 - u{ r k ; 1 < k < y} . Suppose there was X in ["2]"l with [XI3 C rl. Then [XI3 n K l o f 8, so there would be X' in [XIq1 with [X'I3 C K. Since K fl Kol = 8 and PO C K we would have [X'I3 C AT. Then [XI3 nK o l = $, so there would be X ' in [XIq0 with [X'I3 C K. But then we would have [X"I3 C A t , since [X"I3 C ro n Po C A,*. So to prove the theorem, it is enough to show: for every k with k < y,if [XI3 C_ A: then 1x1< q k i1 .

Let Hko be the least of the H k . Then A , , and SO H k o c Hko+l and Hko < H k o + l , SO Hko+l E A,. Thus His not homogeneous. [aHo As a final remark in this section, note that the symbol K -+ ( q k ; k < 7)" can also be defined when K and the 7)k are ordinals, or in fact arbitrary order types. Rather than speaking of the cardinalities of the sets involved, their order type is specified, for a suitable ordering. Some of the results for ordinal numbers will be discussed in Chapter 7. However, in this chapter we shall contine ourselves to the problems involving cardinal numbers.

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