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Denote ∗-finite tuples and A, B, C, . . finite sets of ∗-finite tuples of elements of C. Finite tuples or sets of elements of C will be called standard. Most importantly, in the new set-up Proposition 1 remains valid, also (L) holds in the same way as for standard tuples. Theorem 4 is true, except that for a ∗-finite type p, M(p) may be larger than U (p). Unfortunately Theorem 5 does not hold for ∗-finite types. The change of the set-up does not affect the value of the M-ranks of standard types.

290 (1985), 711–723. [24] , Partitioning pairs of countable ordinals, Acta Mathematica, 159 (1987), 261–294. [25] , Partition problems in topology, Amer. Math. , Providence, 1989. [26] , Irrendundant sets in Boolean algebras, Trans. Amer. Math. , 339 (1993), 35–44. [27] , A classification of transitive relations on ω1 , Proc. London Math. Soc. 73 (1996), 501–533. [28] [29] [30] , Analytic gaps, Fund. Math. 150 (1996), 55–66. A. DiPrisco et al, eds), Kluwer Acad. Press 1997, pp. 213–226. , Compact sets of the first Baire class, preprint 1997.

For example, given two ordinals β > α one can step from β down towards α along the set cβ . More Documenta Mathematica · Extra Volume ICM 1998 · II · 43–52 44 S. Todorcevic precisely, one can define the step from β towards α as the minimal point ξ of cβ such that ξ ≥ α. Let cβ (α), or simply β(α), denote this ordinal. Now one can step further from β(α) towards α and get β(α)(α) (= (β(α))(α)), and so on. This leads us to the notion of a minimal walk from β to α β > β(α) > β(α)(α) > · · · > β(α)(α) · · · (α) = α.

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