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This paper investigates the principles $${\square^{{{\rm ta}}}_{\lambda,\delta}}$$ , weakenings of $${\square_\lambda}$$ which allow $${\delta}$$ many clubs at each level but require them to agree on a tail-end. First, we prove that $${\square^{{\rm {ta}}}_{\lambda,< \omega}}$$ implies $${\square_\lambda}$$ . Then, by forcing from a model with a measurable cardinal, we show that $${\square_{\lambda,2}}$$ does not imply $${\square^{{\rm{ta}}}_{\lambda,\delta}}$$ for regular $${\lambda}$$ , and $${\square^{{\rm{ta}}}_{\delta^+,\delta}}$$ does not imply $${\square_{\delta^+,< \delta}}$$ . With a supercompact cardinal the former result can be extended to singular λ, and the latter can be improved to show that $${\square^{{\rm {ta}}}_{\lambda,\delta}}$$ does not imply $${\square_{\lambda,< \delta}}$$ for $${\delta < \lambda}$$ . [ABSTRACT FROM AUTHOR]

Abstract: We study combinatorial principles weaker than Ramsey’s theorem for pairs over the (recursive comprehension axiom) system with -bounding. It is shown that the cohesiveness (), ascending and descending sequence (), and chain/antichain () principles are all -conservative over -bounding. In particular, none of these principles proves -induction. [Copyright &y& Elsevier]