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By Marco A. Pérez B.

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The Cantor middle thirds set C is a well known self-similar set obtained from I by removing all points whose base 3 representation necessarily includes the digit 1. In other words, the open middle third interval is removed from I, then the open middle thirds are removed from the two obtained intervals, etc. ) In this case C is the disjoint Figure 3. The first three steps in the construction of the middle thirds Cantor set union of the two similar subsets C0 = φ0 (C) and C1 = φ1 (C) where the similarities φ0 and φ1 are given by φ0 (x) = x , 3 φ1 (x) = x 2 + .

It is relatively easy to show that C is a Fra¨ıss´e class. We denote its Fra¨ıss´e limit by QU, and call it the rational Urysohn space. Then U is the completion of QU. Vershik [69] showed the analogue of the Erd˝os–R´enyi theorem, namely, that U is the random Polish space. Of course, the interpretation of this statement requires some care. Suppose that we build a countable metric space by adding one point at a time. At the nth stage, we have points a0 , . . , an−1 , and wish to add a point an with specified distances from the existing points.

621, American Mathematical Society, Providence, RI, 1998. Cameron: Infinite permutation groups 34 [26] J. Covington, H. D. Macpherson and A. D. Mekler, Some maximal subgroups of the symmetric groups, Quart. J. Math. Oxford (2) 47 (1996), 297–311. [27] M. Droste, Structure of partially ordered sets with transitive automorphism groups, Memoirs Amer. Math. Soc. 57 (1985). [28] P. Erd˝ os and A. R´enyi, Asymmetric graphs, Acta Math. Acad. Sci. Hungar. 14 (1963), 295–315. [29] R. Fra¨ıss´e, Sur certains relations qui g´en´eralisent l’ordre des nombres rationnels, C.

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Baer sums by Marco A. Pérez B.

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