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Math. Logic"],"published-print":{"date-parts":[[2023,7]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We study <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\kappa $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03ba<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-maximal cofinitary groups for <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\kappa $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03ba<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> regular uncountable, <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\kappa = \\kappa ^{&lt;\\kappa }$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>\u03ba<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>\u03ba<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mo>&lt;<\/mml:mo>\n                        <mml:mi>\u03ba<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Revisiting earlier work of Kastermans and building upon a recently obtained higher analogue of Bell\u2019s theorem, we show that: <jats:list list-type=\"order\">\n                \n                  \n                  <jats:list-item>\n                    <jats:p>Any <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\kappa $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mi>\u03ba<\/mml:mi>\n                      <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-maximal cofinitary group has <jats:inline-formula><jats:alternatives><jats:tex-math>$${&lt;}\\kappa $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:mo>&lt;<\/mml:mo>\n                          <mml:mi>\u03ba<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:math><\/jats:alternatives><\/jats:inline-formula> many orbits under the natural group action of <jats:inline-formula><jats:alternatives><jats:tex-math>$$S(\\kappa )$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:mi>S<\/mml:mi>\n                          <mml:mo>(<\/mml:mo>\n                          <mml:mi>\u03ba<\/mml:mi>\n                          <mml:mo>)<\/mml:mo>\n                        <\/mml:mrow>\n                      <\/mml:math><\/jats:alternatives><\/jats:inline-formula> on <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\kappa $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mi>\u03ba<\/mml:mi>\n                      <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>\n                  <\/jats:list-item>\n                \n                \n                  \n                  <jats:list-item>\n                    <jats:p>If <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathfrak {p}(\\kappa ) = 2^\\kappa $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:mrow>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03ba<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:mo>=<\/mml:mo>\n                          <mml:msup>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\u03ba<\/mml:mi>\n                          <\/mml:msup>\n                        <\/mml:mrow>\n                      <\/mml:math><\/jats:alternatives><\/jats:inline-formula> then any partition of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\kappa $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mi>\u03ba<\/mml:mi>\n                      <\/mml:math><\/jats:alternatives><\/jats:inline-formula> into less than <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\kappa $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mi>\u03ba<\/mml:mi>\n                      <\/mml:math><\/jats:alternatives><\/jats:inline-formula> many sets can be realized as the orbits of a <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\kappa $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mi>\u03ba<\/mml:mi>\n                      <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-maximal cofinitary group.<\/jats:p>\n                  <\/jats:list-item>\n                \n                \n                  \n                  <jats:list-item>\n                    <jats:p>For any regular <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\lambda &gt; \\kappa $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:mi>\u03bb<\/mml:mi>\n                          <mml:mo>&gt;<\/mml:mo>\n                          <mml:mi>\u03ba<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:math><\/jats:alternatives><\/jats:inline-formula> it is consistent that there is a <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\kappa $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mi>\u03ba<\/mml:mi>\n                      <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-maximal cofinitary group which is universal for groups of size <jats:inline-formula><jats:alternatives><jats:tex-math>$${&lt;}2^\\kappa = \\lambda $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:mo>&lt;<\/mml:mo>\n                          <mml:msup>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\u03ba<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:mo>=<\/mml:mo>\n                          <mml:mi>\u03bb<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. If we only require the group to be universal for groups of size <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\kappa $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mi>\u03ba<\/mml:mi>\n                      <\/mml:math><\/jats:alternatives><\/jats:inline-formula> then this follows from <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathfrak {p}(\\kappa ) = 2^\\kappa $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:mrow>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>\u03ba<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:mo>=<\/mml:mo>\n                          <mml:msup>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>\u03ba<\/mml:mi>\n                          <\/mml:msup>\n                        <\/mml:mrow>\n                      <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.\n<\/jats:p>\n                  <\/jats:list-item>\n                \n              <\/jats:list><\/jats:p>","DOI":"10.1007\/s00153-022-00859-x","type":"journal-article","created":{"date-parts":[[2022,12,4]],"date-time":"2022-12-04T13:02:12Z","timestamp":1670158932000},"page":"641-655","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The structure of $$\\kappa $$-maximal cofinitary groups"],"prefix":"10.1007","volume":"62","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4710-8241","authenticated-orcid":false,"given":"Vera","family":"Fischer","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Corey Bacal","family":"Switzer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,12,4]]},"reference":[{"key":"859_CR1","unstructured":"Andreas, B., Tapani, H.,Yi,\u00a0Z.: Mad families and their neighbors. 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