{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:23:10Z","timestamp":1740108190545,"version":"3.37.3"},"reference-count":8,"publisher":"Springer Science and Business Media LLC","issue":"5-6","license":[{"start":{"date-parts":[[2022,12,16]],"date-time":"2022-12-16T00:00:00Z","timestamp":1671148800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,12,16]],"date-time":"2022-12-16T00:00:00Z","timestamp":1671148800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Swiss Federal Institute of Technology Zurich"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Arch. Math. Logic"],"published-print":{"date-parts":[[2023,7]]},"abstract":"Abstract<\/jats:title>For $$n\\in \\omega $$<\/jats:tex-math>\n \n n<\/mml:mi>\n \u2208<\/mml:mo>\n \u03c9<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the weak choice principle $$\\textrm{RC}_n$$<\/jats:tex-math>\n \n RC<\/mml:mtext>\n n<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is defined as follows:\n For every infinite set<\/jats:italic>X<\/jats:italic>there is an infinite subset<\/jats:italic>$$Y\\subseteq X$$<\/jats:tex-math>\n \n Y<\/mml:mi>\n \u2286<\/mml:mo>\n X<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>with a choice function on<\/jats:italic>$$[Y]^n:=\\{z\\subseteq Y:|z|=n\\}$$<\/jats:tex-math>\n \n \n \n [<\/mml:mo>\n Y<\/mml:mi>\n ]<\/mml:mo>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msup>\n :<\/mml:mo>\n =<\/mml:mo>\n \n {<\/mml:mo>\n z<\/mml:mi>\n \u2286<\/mml:mo>\n Y<\/mml:mi>\n :<\/mml:mo>\n |<\/mml:mo>\n z<\/mml:mi>\n |<\/mml:mo>\n =<\/mml:mo>\n n<\/mml:mi>\n }<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>\n <\/jats:disp-quote>The choice principle $$\\textrm{C}_n^-$$<\/jats:tex-math>\n \n C<\/mml:mtext>\n n<\/mml:mi>\n -<\/mml:mo>\n <\/mml:msubsup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> states the following:\n For every infinite family of<\/jats:italic>n<\/jats:italic>-element sets, there is an infinite subfamily<\/jats:italic>$${\\mathcal {G}}\\subseteq {\\mathcal {F}}$$<\/jats:tex-math>\n \n G<\/mml:mi>\n \u2286<\/mml:mo>\n F<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>with a choice function.<\/jats:italic><\/jats:p>\n <\/jats:disp-quote>The choice principles $$\\textrm{LOC}_n^-$$<\/jats:tex-math>\n \n LOC<\/mml:mtext>\n n<\/mml:mi>\n -<\/mml:mo>\n <\/mml:msubsup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and $$\\textrm{WOC}_n^-$$<\/jats:tex-math>\n \n WOC<\/mml:mtext>\n n<\/mml:mi>\n -<\/mml:mo>\n <\/mml:msubsup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are the same as $$\\textrm{C}_n^-$$<\/jats:tex-math>\n \n C<\/mml:mtext>\n n<\/mml:mi>\n -<\/mml:mo>\n <\/mml:msubsup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, but we assume that the family $${\\mathcal {F}}$$<\/jats:tex-math>\n F<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is linearly orderable (for $$\\textrm{LOC}_n^-$$<\/jats:tex-math>\n \n LOC<\/mml:mtext>\n n<\/mml:mi>\n -<\/mml:mo>\n <\/mml:msubsup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) or well-orderable (for $$\\textrm{WOC}_n^-$$<\/jats:tex-math>\n \n WOC<\/mml:mtext>\n n<\/mml:mi>\n -<\/mml:mo>\n <\/mml:msubsup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>). In the first part of this paper, for $$m,n\\in \\omega $$<\/jats:tex-math>\n \n m<\/mml:mi>\n ,<\/mml:mo>\n n<\/mml:mi>\n \u2208<\/mml:mo>\n \u03c9<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> we will give a full characterization of when the implication $$\\textrm{RC}_m\\Rightarrow \\textrm{WOC}_n^-$$<\/jats:tex-math>\n \n \n RC<\/mml:mtext>\n m<\/mml:mi>\n <\/mml:msub>\n \u21d2<\/mml:mo>\n \n WOC<\/mml:mtext>\n n<\/mml:mi>\n -<\/mml:mo>\n <\/mml:msubsup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> holds in $${\\textsf {ZF}}$$<\/jats:tex-math>\n ZF<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We will prove the independence results by using suitable Fraenkel-Mostowski permutation models. In the second part, we will show some generalizations. In particular, we will show that $$\\textrm{RC}_5\\Rightarrow \\textrm{LOC}_5^-$$<\/jats:tex-math>\n \n \n RC<\/mml:mtext>\n 5<\/mml:mn>\n <\/mml:msub>\n \u21d2<\/mml:mo>\n \n LOC<\/mml:mtext>\n 5<\/mml:mn>\n -<\/mml:mo>\n <\/mml:msubsup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and that $$\\textrm{RC}_6\\Rightarrow \\textrm{C}_3^-$$<\/jats:tex-math>\n \n \n RC<\/mml:mtext>\n 6<\/mml:mn>\n <\/mml:msub>\n \u21d2<\/mml:mo>\n \n C<\/mml:mtext>\n 3<\/mml:mn>\n -<\/mml:mo>\n <\/mml:msubsup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, answering two open questions from Halbeisen and Tachtsis (Arch Math Logik 59(5):583\u2013606, 2020). Furthermore, we will show that $$\\textrm{RC}_6\\Rightarrow \\textrm{C}_9^-$$<\/jats:tex-math>\n \n \n RC<\/mml:mtext>\n 6<\/mml:mn>\n <\/mml:msub>\n \u21d2<\/mml:mo>\n \n C<\/mml:mtext>\n 9<\/mml:mn>\n -<\/mml:mo>\n <\/mml:msubsup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and that $$\\textrm{RC}_7\\Rightarrow \\textrm{LOC}_7^-$$<\/jats:tex-math>\n \n \n RC<\/mml:mtext>\n 7<\/mml:mn>\n <\/mml:msub>\n \u21d2<\/mml:mo>\n \n LOC<\/mml:mtext>\n 7<\/mml:mn>\n -<\/mml:mo>\n <\/mml:msubsup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.\n<\/jats:p>","DOI":"10.1007\/s00153-022-00860-4","type":"journal-article","created":{"date-parts":[[2022,12,16]],"date-time":"2022-12-16T20:20:32Z","timestamp":1671222032000},"page":"703-733","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Some implications of Ramsey Choice for families of $$\\varvec{n}$$-element sets"],"prefix":"10.1007","volume":"62","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6078-7237","authenticated-orcid":false,"given":"Lorenz","family":"Halbeisen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Salome","family":"Schumacher","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,12,16]]},"reference":[{"issue":"3","key":"860_CR1","doi-asserted-by":"publisher","first-page":"867","DOI":"10.1090\/S0002-9939-98-04103-3","volume":"126","author":"O De La Cruz","year":"1998","unstructured":"De La Cruz, O., Di Prisco, C.A.: Weak choice principles. Proc. Am. Math. Soc. 126(3), 867\u2013876 (1998)","journal-title":"Proc. Am. Math. Soc."},{"key":"860_CR2","doi-asserted-by":"crossref","unstructured":"Halbeisen, L., Hungerb\u00fchler, N., Lazarovich, N., Lederle, W., Lischka, M., Schumacher, S.: Forms of choice in ring theory. Results Math. 74(1), Article:14 (2019)","DOI":"10.1007\/s00025-018-0935-1"},{"key":"860_CR3","unstructured":"Halbeisen, L., Plati, R., Schumacher, S.: A new weak choice principle. arXiv:2101.07840"},{"issue":"5","key":"860_CR4","doi-asserted-by":"publisher","first-page":"583","DOI":"10.1007\/s00153-019-00705-7","volume":"59","author":"L Halbeisen","year":"2020","unstructured":"Halbeisen, L., Tachtsis, E.: On Ramsey Choice and partial choice for infinite families of $$n$$-element sets. Arch. Math. Logik 59(5), 583\u2013606 (2020)","journal-title":"Arch. Math. Logik"},{"key":"860_CR5","volume-title":"Combinatorial Set Theory\u2014With a Gentle Introduction to Forcing","author":"LJ Halbeisen","year":"2012","unstructured":"Halbeisen, L.J.: Combinatorial Set Theory\u2014With a Gentle Introduction to Forcing. Springer, London (2012)"},{"key":"860_CR6","unstructured":"Montenegro, C.H.: Weak versions of the axiom of choice for families of finite sets. In: Caicedo, X., Montenegro, C. (Eds.) Models, Algebras, and Proofs, Selected Papers of the X Latin American Symposium on Mathematical Logic, Bogot\u00e0, Colombia, June 24\u201329, 1995, Lecture Notes in Pure and Applied Mathematics, vol. 203, pp. 57\u201360. Dekker, New York (1999)"},{"key":"860_CR7","doi-asserted-by":"publisher","first-page":"721","DOI":"10.2307\/2272420","volume":"37","author":"D Pincus","year":"1972","unstructured":"Pincus, D.: Zermelo\u2013Fraenkel consistency results by Fraenkel\u2013Mostowski methods. J. Symb. Logic 37, 721\u2013743 (1972)","journal-title":"J. Symb. 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