We investigate uncountable maximal antichains of perfect trees and of splitting trees. We show that in the case of perfect trees they must have size of at least the dominating number, whereas for splitting trees they are of size at least \(\mathsf {cov}(\mathcal {M})\), i.e. the covering coefficient of the meager ideal. Finally, we show that uncountable maximal antichains of superperfect trees are at least of size the bounding number; moreover we show that this is best possible.

中文翻译：

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完美和分裂树木的反链

我们研究了完美树和分裂树的无数最大反链。我们表明，对于完美的树木，它们的大小必须至少为主导数，而对于分裂树木，它们的大小至少为\（\ mathsf {cov}（\ mathcal {M}）\），即覆盖理想的系数。最后，我们证明了超完美树的不可数的最大反链至少具有边界数的大小。此外，我们证明这是最好的可能。