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The uniform Kruskal theorem extends the original result for trees to generalrecursive data types. As shown by A. Freund, M. Rathjen and A. Weiermann, it isequivalent to $\Pi^1_1$-comprehension, over $\mathsf{RCA_0}$ with the chainantichain principle ($\mathsf{CAC}$). This result provides a connection betweenfinite combinatorics and abstract set existence. The present paper shedsfurther light on this connection. First, we show that the original Kruskaltheorem is equivalent to the uniform version for data types that are finitelygenerated. Secondly, we prove a dichotomy result for a natural variant of theuniform Kruskal theorem. On the one hand, this variant still implies$\Pi^1_1$-comprehension over $\mathsf{RCA}_0+\mathsf{CAC}$. On the other hand,it becomes weak when $\mathsf{CAC}$ is removed from the base theory.

The uniform Kruskal theorem: between finite combinatorics and strong set existence