It has been known since at least [ER52] that infinite-exponent partition relations (IEPRs) are inconsistent with AC, i.e. that no higher analogue of Ramsey's theorem can hold for colourings of infinite tuples; in ZF without AC, however, such relations can hold. The same is true of structural IEPRs, i.e. ones in which the "tuples" being coloured are the isomorphic copies of some given structure inside a larger structure. In this talk we will discuss some particular structural IEPRs which are consistent with ZF but which imply failures of fragments of Choice such as the Kinna-Wagner Selection Principle and the Ordering Principle.
This is joint work with Jonathan Schilhan, and appears in [GS26].
References
[ER52] P. Erdős, R. Rado, Combinatorial theorems on classifications of subsets of a given set, Proceedings of the London Mathematical Society, Volume s3–2 (1952), pp. 417–439
[GS26] L. A. Gardiner, J. Schilhan, Structural infinite-exponent partition relations and weak choice principles, arXiv:2605.21259 (2026)