Science Mathematics

# The Heart of Mathematics by P. R. Halmos

By P. R. Halmos

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Example text

G. satisfies with with A'^ = B1 oo-homogeneous by f a | A , = i//1 , and a LA = ij; and A = B. 30 MANFRED DROSTE Now let r + |lP(ft)| + 1 £ m £ U. We claim that neous. Let D,E c ft with Since 1 |D'| > r+1, D D'^ = E' A(ft) is not an antichain, thus connected. We obtain A(ft) is m-homogeneous and is m-transitive, m = r + |lP(ft)| an antichain with 7T £ A(F 0 G) is m-homoge- |D| = |E| = m and ij;: D -»• E an isomorphism. and (D°)^ = E°. As before there exists Finally, suppose A(ft) 3 € A(ft) with B| = ip.

We claim that A a = B for some suffices to find two finite connected subsets \p: A -*• B e3d c ft \ ft. all A(Q) G ft and either (1) phism iso- properties: onto Whenever (i) A(ft) following Whenever for (3) with B = {inf T; T c B} be a tree. ft A = [a3o]Q+ (Hi) to an cp 3-transitive. 33. B c ft3 B. Clearly, if A,B

G. Put c •< h a < d = {c,d}. Thus ram(ft) (d) h = inf{f,g}. Then ft and h a £ ram (ft). So a £ A(ft) ram(ft) is is maximal or minimal, this implies is unbounded in ft. A c ft. Choose Suppose there were a finite maximal antichain a £ ft with a < A with and c || a a < c < d < b. e £ h < f. 2(b) there exists dense in ft. Since no element of that c,d € ft with b £ ram(ft) and with b = inf{a,c}. Then b < a. There exists c || A and thus c € ft A 0 {c} <= ft is an antichain, a contradiction. The following theorem contains a first important result.