By D. G. Higman (auth.), Derek A. Holton (eds.)

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**Extra info for Combinatorial Mathematics: Proceedings of the Second Australian Conference**

**Example text**

C. Bessaga) Let X be a set and f: X§ ates fn have a unique fixed point. be a map such that all the iter- Assuming the weak (countable) form of the axiom of choice, then for any c e (0,i) there exists a complete metric on X rendering f a c-contraction. The purpose of this note is to show that in case X has at most continuously many elements then the separability of the metric in the above theorem can be claimed. P. Ramsey [3]. THEOREM 2. P. Ramsey) N is decomposed If the set of all unordered pairs {n,m} of natural numbers in finite number of sets, say R I , R 2 , .

The relation established in Theorem 8 between A and its um-pi looks the same as the relation between an element and its inverse in the We also remark that the converse of Theorem 8 EXAMPLE 2. A special linear ~roup. is not true. Let be an e l e m e n t o f But B(3). A= 1 0 A+ : 1 0 Then = com A. is a singular matrix. The p r e c e d i n g example s u g g e s t s t h e f o l l o w i n g p r o p o s i t i o n ing, but not necessary for understanding the material PROPOSITION 9. Then I__nnparticular, A Let is contained in Proof.

The trees Pm' m ~ h, and E 7 do not fall into the class of trees considered. Theorem i0, T is semi-stable at an endvertex w. Tw may be expressed in the form (P2n+l Thus T = (P2n+l ; c)-(M ; c)'(Mw As T is not (P2n+l By ; c) - determined, ; z) where w is an endvertex of M. ; z), and T is semi-stable at the endvertex w lying in M #. This completes the proof. We define as (2n + i) - peculiar those trees other than P2n+l which are (P2n+l ; c)-determined and which, when expressed as (P2n+l ; c)'(H ; v) are not semi- stable at an endvertex lying in H #.