Garti 2019 first omitting cardinal magidority

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claim_1_10: For every successor ordinal beta it is consistent, from large cardinals, that some Magidor cardinal lambda has alpha_M(lambda) = aleph_{beta+1}, and it is consistent that alpha_M(lambda) is the successor of a strongly inaccessible, even strongly Mahlo, cardinal.

theorem_1_12: It is consistent that lambda is Magidor and alpha_M(lambda) = mu^+ with mu supercompact, answering positively the authors' earlier question whether alpha_M can be the successor of a measurable cardinal.

theorem_1_2: For a Magidor cardinal lambda, alpha_M is a successor cardinal when no Magidor cardinal lies in the interval from alpha_M to 2^{alpha_M}, and in particular for every Magidor cardinal when every limit cardinal is a strong limit.

theorem_1_4: If lambda is Magidor, kappa < lambda is measurable with 2^kappa < lambda, and lambda stays Magidor after Prikry forcing through a normal ultrafilter on kappa, then in the extension alpha_M exceeds kappa^omega and so exceeds kappa^+.

theorem_2_7: If lambda is I1, one can force alpha_M(lambda) = mu^+ with mu a singular cardinal of uncountable cofinality, by Magidor forcing over a supercompact mu with alpha_M^{<mu}(lambda) = mu^+.

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Shimon Garti, Yair Hayut, The first omitting cardinal for Magidority. Mathematical Logic Quarterly 65 (2019), no. 1, 95--104. arXiv:1801.00239, doi:10.1002/malq.201800026. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1801.00239), every other right reserved.

The copy read for this card is arXiv version 3 (16 May 2019), and the page numbers below are its own. A cardinal lambda is Magidor when lambda -> [lambda]^{aleph_0-bd}_lambda, a relation for colorings of the countable bounded subsets of lambda, and alpha_M(lambda) is the least alpha < lambda with lambda -> [lambda]^{aleph_0-bd}_alpha. Theorem 1.12 (p. 11) shows that it is consistent that lambda is Magidor and alpha_M(lambda) = mu^+ with mu supercompact, and Theorem 2.7 (p. 16) shows that from an I1 cardinal lambda one can force alpha_M(lambda) = mu^+ with mu singular of uncountable cofinality. The introduction (p. 2) explains the restriction to bounded subsets by a theorem of Erdős and Hajnal: lambda -/-> [lambda]^{aleph_0}_lambda for every infinite cardinal lambda. The method is set-theoretic forcing and large-cardinal consistency arguments. The arXiv identifier 1801.00239 identifies the Garti-Hayut paper recorded here, not 'On a problem of Erdos and Hajnal' with Shelah as a coauthor (garti_2025_problem_erdos_hajnal). It was consulted for problem 598 because a reply on the problem's discussion thread (post 4801, 2026-03-15) reported that Zeraoulia Rafik's partial result there follows from work of Garti and Hayut, and because the claim pages of problem 598 cite its Claim 1.10(a).

Read status: claims checked for the abstract and introduction (pp. 1--3), Theorem 1.2 with Lemma 1.1 (pp. 4--5), Theorem 1.4 (p. 6), Lemma 1.9 and Claim 1.10 (pp. 9--10), Definition 1.11 and Theorem 1.12 (p. 11), Conjecture 2.1 and Definition 2.2 (p. 13), Claim 2.4 (p. 14), Claim 2.5 (p. 15), Lemma 2.6 and Theorem 2.7 (p. 16) and Question 2.8 (p. 18), each read clause by clause on the printed pages of arXiv v3. The proofs were read for structure only, and nothing here is independently reviewed.

Results.

• Theorem 1.2 (p. 5): for Magidor lambda, alpha_M is a successor cardinal if no Magidor cardinal lies in [alpha_M, 2^{alpha_M}], hence for every Magidor lambda if every limit cardinal is a strong limit.
• Theorem 1.4 (p. 6): Prikry forcing at a measurable kappa with 2^kappa < lambda, if it keeps lambda Magidor, gives alpha_M > (kappa^omega)^{V[G]}.
• Claim 1.10 (p. 10), with Lemma 1.9 (p. 9): for every successor ordinal beta it is consistent from large cardinals that alpha_M(lambda) = aleph_{beta+1} for some Magidor lambda, and consistent that alpha_M(lambda) is the successor of a strongly inaccessible, even strongly Mahlo, cardinal.
• Theorem 1.12 (p. 11): it is consistent that lambda is Magidor and alpha_M(lambda) = mu^+ with mu supercompact.
• Theorem 2.7 (p. 16): from an I1 cardinal lambda one can force alpha_M(lambda) = mu^+ with mu singular of uncountable cofinality.

Source: https://arxiv.org/abs/1801.00239.

Bears on. #598: the paper does not mention the problem. The introduction (p. 2) recalls, as a theorem of Erdős and Hajnal, that lambda -/-> [lambda]^{aleph_0}_lambda for every infinite cardinal lambda: some coloring of the countable subsets of lambda with lambda colors gives every subset of size lambda countable subsets of every color; at lambda = (2^{aleph_0})^+ this is the case m = (2^{aleph_0})^+ of #598. Claim 1.10(a) (p. 10), at beta = 1, gives the consistency, from large cardinals, of a Magidor cardinal lambda with alpha_M(lambda) = aleph_2, a statement about colorings of the countable bounded subsets of lambda; two claim pages of #598 cite it as the source of their model, and the paper draws no consequence for #598 and does not state the value of 2^{aleph_0} in that model.

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