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Later work by S\u00e1rk\u00f6zy, Selkow and Szemer\u00e9di showed that in fact Dirac graphs have\n                    <jats:italic>many<\/jats:italic>\n                    Hamilton cycles and perfect matchings, culminating in a result of Cuckler and Kahn that gives a precise description of the numbers of Hamilton cycles and perfect matchings in a Dirac graph\n                    <jats:italic>G<\/jats:italic>\n                    (in terms of an entropy-like parameter of\n                    <jats:italic>G<\/jats:italic>\n                    ). In this paper we extend Cuckler and Kahn\u2019s result to perfect matchings in hypergraphs. For positive integers\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$d&lt;k$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>d<\/mml:mi>\n                            <mml:mo>&lt;<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , and for\n                    <jats:italic>n<\/jats:italic>\n                    divisible by\n                    <jats:italic>k<\/jats:italic>\n                    , let\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$m_{d}(k,n)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>m<\/mml:mi>\n                              <mml:mi>d<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>k<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    be the minimum\n                    <jats:italic>d<\/jats:italic>\n                    -degree that ensures the existence of a perfect matching in an\n                    <jats:italic>n<\/jats:italic>\n                    -vertex\n                    <jats:italic>k<\/jats:italic>\n                    -uniform hypergraph. In general, it is an open question to determine (even asymptotically) the values of\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$m_{d}(k,n)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>m<\/mml:mi>\n                              <mml:mi>d<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>k<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , but we are nonetheless able to prove an analogue of the Cuckler\u2013Kahn theorem, showing that if an\n                    <jats:italic>n<\/jats:italic>\n                    -vertex\n                    <jats:italic>k<\/jats:italic>\n                    -uniform hypergraph\n                    <jats:italic>G<\/jats:italic>\n                    has minimum\n                    <jats:italic>d<\/jats:italic>\n                    -degree at least\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$(1+\\gamma )m_{d}(k,n)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mi>\u03b3<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:msub>\n                              <mml:mi>m<\/mml:mi>\n                              <mml:mi>d<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>k<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    (for any constant\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\gamma &gt;0$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>\u03b3<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    ), then the number of perfect matchings in\n                    <jats:italic>G<\/jats:italic>\n                    is controlled by an entropy-like parameter of\n                    <jats:italic>G<\/jats:italic>\n                    . This strengthens cruder estimates arising from work of Kang\u2013Kelly\u2013K\u00fchn\u2013Osthus\u2013Pfenninger and Pham\u2013Sah\u2013Sawhney\u2013Simkin.\n                  <\/jats:p>","DOI":"10.1007\/s00493-025-00194-8","type":"journal-article","created":{"date-parts":[[2026,1,28]],"date-time":"2026-01-28T10:15:41Z","timestamp":1769595341000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Counting Perfect Matchings in Dirac Hypergraphs"],"prefix":"10.1007","volume":"46","author":[{"given":"Matthew","family":"Kwan","sequence":"first","affiliation":[]},{"given":"Roodabeh","family":"Safavi","sequence":"additional","affiliation":[]},{"given":"Yiting","family":"Wang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,1,28]]},"reference":[{"key":"194_CR1","unstructured":"Bondy, J.A.: Basic graph theory: paths and circuits, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. 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