Let κ be a cardinal which is measurable after generically adding κ+ω many Cohen subsets to κ, and let Qκ = (Q, ≤ Q ) be the strongly κ-dense linear order of size κ. We prove, for 2 ≤ m < ω, that there is a finite value t + m such that the set [Q] m of m-tuples from Q can be partitioned into classes C i : i < t + m such that for any coloring a class C i in fewer thanthat is, for any coloring of [Qκ] m with fewer than κ colors there is a copy Q ⊆ Q of Qκ such that [Q ] m has at most t + m colors. On the other hand, we show that there are colorings of Qκ such that if Q ⊆ Q is any copy of Qκ then C i ∩ [Q ] = ∅ for all i < t + m , and hence Qκ [Qκ] m t + m .We characterize t + m as the cardinality of a certain finite set of ordered trees and obtain an upper and a lower bound on its value. In particular, t + 2 = 2 and for m > 2 we have t + m > tm, the m-th tangent number. The stated condition on κ is the hypothesis for a result of Shelah on which our work relies. A model in which this condition holds simultaneously for all m can be obtained by forcing from a model with a κ +ω -strong cardinal, but it follows from earlier results of Hajnal and Komjáth that our result, and hence Shelah's theorem, is not directly implied by any large cardinal assumption.
Joyce trees have concrete realizations as J-trees of sequences of 0's and 1's. Algorithms are given for computing the number of minimal height J-trees of d-ary sequences with n leaves and the number of them with minimal parent passing numbers to obtain polynomials ρn(d) for the full collection and αn(d) for the subcollection.The number of traditional Joyce trees is the tangent number αn(1); αn(2) is the number of cells in the canonical partition by Laflamme, Sauer and Vuksanovic of n-element subsets of the infinite random (Rado) graph; and ρn(2) is the number of weak embedding types of rooted n-leaf J-trees of sequences of 0's and 1's.
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