{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,10,8]],"date-time":"2024-10-08T04:14:47Z","timestamp":1728360887235},"reference-count":3,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2024,5,14]],"date-time":"2024-05-14T00:00:00Z","timestamp":1715644800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,5,14]],"date-time":"2024-05-14T00:00:00Z","timestamp":1715644800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Combinatorica"],"published-print":{"date-parts":[[2024,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>A family <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {F} \\subset \\mathcal {P}(n)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>F<\/mml:mi>\n                    <mml:mo>\u2282<\/mml:mo>\n                    <mml:mi>P<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is <jats:italic>r<\/jats:italic>-<jats:italic>wise<\/jats:italic><jats:italic>k<\/jats:italic>-<jats:italic>intersecting<\/jats:italic> if <jats:inline-formula><jats:alternatives><jats:tex-math>$$|A_1 \\cap \\dots \\cap A_r| \\ge k$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:msub>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>\u2229<\/mml:mo>\n                    <mml:mo>\u22ef<\/mml:mo>\n                    <mml:mo>\u2229<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mi>r<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                      <mml:mo>\u2265<\/mml:mo>\n                      <mml:mi>k<\/mml:mi>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for any <jats:inline-formula><jats:alternatives><jats:tex-math>$$A_1, \\dots , A_r \\in \\mathcal {F}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>\u22ef<\/mml:mo>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mi>r<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mi>F<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. It is easily seen that if <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {F}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>F<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is <jats:italic>r<\/jats:italic>-wise <jats:italic>k<\/jats:italic>-intersecting for <jats:inline-formula><jats:alternatives><jats:tex-math>$$r \\ge 2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>r<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$k \\ge 1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> then <jats:inline-formula><jats:alternatives><jats:tex-math>$$|\\mathcal {F}| \\le 2^{n-1}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                      <mml:mi>F<\/mml:mi>\n                      <mml:mo>|<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:msup>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mrow>\n                        <mml:mi>n<\/mml:mi>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The problem of determining the maximum size of a family <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {F}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>F<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that is both <jats:inline-formula><jats:alternatives><jats:tex-math>$$r_1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>r<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-wise <jats:inline-formula><jats:alternatives><jats:tex-math>$$k_1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-intersecting and <jats:inline-formula><jats:alternatives><jats:tex-math>$$r_2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>r<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-wise <jats:inline-formula><jats:alternatives><jats:tex-math>$$k_2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-intersecting was raised in 2019 by Frankl and Kupavskii (Combinatorica 39:1255\u20131266, 2019). They proved the surprising result that, for <jats:inline-formula><jats:alternatives><jats:tex-math>$$(r_1,k_1) = (3,1)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>r<\/mml:mi>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>k<\/mml:mi>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mn>3<\/mml:mn>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mn>1<\/mml:mn>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$(r_2,k_2) = (2,32)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>r<\/mml:mi>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>k<\/mml:mi>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mn>32<\/mml:mn>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> then this maximum is at most <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^{n-2}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mrow>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:mrow>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and conjectured the same holds if <jats:inline-formula><jats:alternatives><jats:tex-math>$$k_2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is replaced by 3. In this paper we shall not only prove this conjecture but we shall also determine the exact maximum for <jats:inline-formula><jats:alternatives><jats:tex-math>$$(r_1,k_1) = (3,1)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>r<\/mml:mi>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>k<\/mml:mi>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mn>3<\/mml:mn>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mn>1<\/mml:mn>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$(r_2,k_2) = (2,3)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>r<\/mml:mi>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>k<\/mml:mi>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mn>3<\/mml:mn>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> for all <jats:italic>n<\/jats:italic>.<\/jats:p>","DOI":"10.1007\/s00493-024-00105-3","type":"journal-article","created":{"date-parts":[[2024,5,14]],"date-time":"2024-05-14T10:01:43Z","timestamp":1715680903000},"page":"1053-1061","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Proof of a Frankl\u2013Kupavskii Conjecture on Intersecting Families"],"prefix":"10.1007","volume":"44","author":[{"given":"Agnijo","family":"Banerjee","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,5,14]]},"reference":[{"key":"105_CR1","doi-asserted-by":"publisher","first-page":"1255","DOI":"10.1007\/s00493-019-4064-6","volume":"39","author":"P Frankl","year":"2019","unstructured":"Frankl, P., Kupavskii, A.: Incompatible intersection properties. Combinatorica 39, 1255\u20131266 (2019)","journal-title":"Combinatorica"},{"issue":"2","key":"105_CR2","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1006\/eujc.1995.0092","volume":"18","author":"R Ahlswede","year":"1997","unstructured":"Ahlswede, R., Khachatrian, L.H.: The complete intersection theorem for systems of finite sets. Eur. J. Comb. 18(2), 125\u2013136 (1997)","journal-title":"Eur. J. Comb."},{"key":"105_CR3","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1007\/BF01897141","volume":"15","author":"GOH Katona","year":"1964","unstructured":"Katona, G.O.H.: Intersection theorems for systems of finite sets. Acta Math. Acad. Sci. Hungar. 15, 329\u2013337 (1964)","journal-title":"Acta Math. Acad. Sci. Hungar."}],"container-title":["Combinatorica"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-024-00105-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00493-024-00105-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00493-024-00105-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,7]],"date-time":"2024-10-07T12:08:38Z","timestamp":1728302918000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00493-024-00105-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,14]]},"references-count":3,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2024,10]]}},"alternative-id":["105"],"URL":"https:\/\/doi.org\/10.1007\/s00493-024-00105-3","relation":{},"ISSN":["0209-9683","1439-6912"],"issn-type":[{"type":"print","value":"0209-9683"},{"type":"electronic","value":"1439-6912"}],"subject":[],"published":{"date-parts":[[2024,5,14]]},"assertion":[{"value":"9 May 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 November 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 November 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 May 2024","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}