{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T03:21:31Z","timestamp":1765164091260,"version":"3.46.0"},"reference-count":12,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T00:00:00Z","timestamp":1761782400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T00:00:00Z","timestamp":1761782400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100011019","name":"Nemzeti Kutat\u00e1si Fejleszt\u00e9si \u00e9s Innov\u00e1ci\u00f3s Hivatal","doi-asserted-by":"publisher","award":["K 116769","KH 130371","SNN 129364"],"award-info":[{"award-number":["K 116769","KH 130371","SNN 129364"]}],"id":[{"id":"10.13039\/501100011019","id-type":"DOI","asserted-by":"publisher"}]},{"name":"HUN-REN Alfr\u00e9d R\u00e9nyi Institute of Mathematics"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Graphs and Combinatorics"],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Bern\u00e1th and Gerbner in 2007 introduced (\n                    <jats:italic>p<\/jats:italic>\n                    ,\u00a0\n                    <jats:italic>q<\/jats:italic>\n                    )-chain intersecting families of subsets of an\n                    <jats:italic>n<\/jats:italic>\n                    -element underlying set. Those have the property that for any\n                    <jats:italic>p<\/jats:italic>\n                    -chain\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$A_1\\subsetneq A_2\\subsetneq \\dots \\subsetneq A_p$$<\/jats:tex-math>\n                        <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>\u228a<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>A<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>\u228a<\/mml:mo>\n                            <mml:mo>\u22ef<\/mml:mo>\n                            <mml:mo>\u228a<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>A<\/mml:mi>\n                              <mml:mi>p<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:italic>q<\/jats:italic>\n                    -chain\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$B_1\\subsetneq B_2\\subsetneq \\dots \\subsetneq B_q$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>\u228a<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>\u228a<\/mml:mo>\n                            <mml:mo>\u22ef<\/mml:mo>\n                            <mml:mo>\u228a<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mi>q<\/mml:mi>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , we have\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$A_p\\cap B_q\\ne \\emptyset $$<\/jats:tex-math>\n                        <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:mi>p<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\u2229<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mi>q<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\u2260<\/mml:mo>\n                            <mml:mi>\u2205<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . Bern\u00e1th and Gerbner determined the largest cardinality of such families. They also introduced strongly (\n                    <jats:italic>p<\/jats:italic>\n                    ,\u00a0\n                    <jats:italic>q<\/jats:italic>\n                    )-chain intersecting families, where\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$A_p\\cap B_1\\ne \\emptyset $$<\/jats:tex-math>\n                        <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:mi>p<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\u2229<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>\u2260<\/mml:mo>\n                            <mml:mi>\u2205<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and totally (\n                    <jats:italic>p<\/jats:italic>\n                    ,\u00a0\n                    <jats:italic>q<\/jats:italic>\n                    )-chain intersecting families, where\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$A_1\\cap B_1\\ne \\emptyset $$<\/jats:tex-math>\n                        <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>\u2229<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>\u2260<\/mml:mo>\n                            <mml:mi>\u2205<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . They obtained some partial results on the maximum cardinality of such families. We extend those results by determining the largest cardinality of strongly (\n                    <jats:italic>p<\/jats:italic>\n                    ,\u00a0\n                    <jats:italic>q<\/jats:italic>\n                    )-chain intersecting families if\n                    <jats:italic>n<\/jats:italic>\n                    is sufficiently large, and by determining the largest cardinality of totally (2,\u00a02)-chain intersecting families.\n                  <\/jats:p>","DOI":"10.1007\/s00373-025-02989-4","type":"journal-article","created":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T02:44:08Z","timestamp":1761792248000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A note on strongly and totally chain intersecting families"],"prefix":"10.1007","volume":"41","author":[{"given":"D\u00e1niel","family":"Gerbner","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,10,30]]},"reference":[{"key":"2989_CR1","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1007\/s00373-007-0743-y","volume":"23","author":"A Bern\u00e1th","year":"2007","unstructured":"Bern\u00e1th, A., Gerbner, D.: Chain intersecting families. Graphs Combin. 23, 353\u2013366 (2007)","journal-title":"Graphs Combin."},{"key":"2989_CR2","doi-asserted-by":"publisher","first-page":"898","DOI":"10.1090\/S0002-9904-1945-08454-7","volume":"51","author":"P Erd\u0151s","year":"1945","unstructured":"Erd\u0151s, P.: On a lemma of littlewood and offord. Bull. Am. Math. Soc. 51, 898\u2013902 (1945)","journal-title":"Bull. Am. Math. Soc."},{"key":"2989_CR3","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1093\/qmath\/12.1.313","volume":"12","author":"P Erd\u0151s","year":"1961","unstructured":"Erd\u0151s, P., Ko, C., Rado, R.: Intersection theorems for systems of finite sets. Quart. J. Math. Oxford 12, 313\u2013320 (1961)","journal-title":"Quart. J. Math. Oxford"},{"key":"2989_CR4","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1137\/0403031","volume":"3","author":"P Frankl","year":"1990","unstructured":"Frankl, P.: Canonical antichains on the circle and applications. SIAM J. Discrete Math. 3, 355\u2013363 (1990)","journal-title":"SIAM J. Discrete Math."},{"key":"2989_CR5","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1007\/s00493-013-2917-y","volume":"33","author":"D Gerbner","year":"2013","unstructured":"Gerbner, D.: Profile polytopes of some classes of families. Combinatorica 33, 199\u2013216 (2013)","journal-title":"Combinatorica"},{"key":"2989_CR6","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1016\/j.jcta.2017.04.009","volume":"151","author":"D Gerbner","year":"2017","unstructured":"Gerbner, D., Methuku, A., Tompkins, C.: Intersecting P-free families. Comb. Theory, Ser. A. 151, 61\u201383 (2017)","journal-title":"Comb. Theory, Ser. A."},{"key":"2989_CR7","doi-asserted-by":"publisher","DOI":"10.1201\/9780429440809","volume-title":"Extremal Finite Set Theory","author":"D Gerbner","year":"2018","unstructured":"Gerbner, D., Patk\u00f3s, B.: Extremal Finite Set Theory, 1st edn. CRC Press, Boca Raton (2018)","edition":"1"},{"key":"2989_CR8","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1093\/qmath\/27.1.33","volume":"27","author":"AJW Hilton","year":"1976","unstructured":"Hilton, A.J.W.: A theorem on finite sets. Quart. J. Math. (Oxford)(2) 27, 33\u201336 (1976)","journal-title":"Quart. J. Math. (Oxford)(2)"},{"key":"2989_CR9","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1016\/0095-8956(72)90054-8","volume":"13","author":"GOH Katona","year":"1972","unstructured":"Katona, G.O.H.: A simple proof of the Erd\u0151s-Chao Ko-Rado theorem. J. Comb. Theory Ser. B 13, 183\u2013184 (1972)","journal-title":"J. Comb. Theory Ser. B"},{"issue":"2","key":"2989_CR10","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1016\/S0021-9800(66)80035-2","volume":"1","author":"D Lubell","year":"1966","unstructured":"Lubell, D.: A short proof of Sperner\u2019s lemma. J. Comb. Theory 1(2), 299 (1966)","journal-title":"J. Comb. Theory"},{"key":"2989_CR11","doi-asserted-by":"publisher","first-page":"204","DOI":"10.1112\/jlms\/s1-43.1.204","volume":"43","author":"EC Milner","year":"1968","unstructured":"Milner, E.C.: A combinatorial theorem on systems of sets. J. London Math. Soc. 43, 204\u2013206 (1968)","journal-title":"J. London Math. Soc."},{"key":"2989_CR12","doi-asserted-by":"publisher","first-page":"544","DOI":"10.1007\/BF01171114","volume":"27","author":"E Sperner","year":"1928","unstructured":"Sperner, E.: Ein Satz \u00fcber Untermengen einer endlichen Menge. Math. Z. 27, 544\u2013548 (1928)","journal-title":"Math. Z."}],"container-title":["Graphs and Combinatorics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-025-02989-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00373-025-02989-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00373-025-02989-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T03:20:16Z","timestamp":1765164016000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00373-025-02989-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,30]]},"references-count":12,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["2989"],"URL":"https:\/\/doi.org\/10.1007\/s00373-025-02989-4","relation":{},"ISSN":["0911-0119","1435-5914"],"issn-type":[{"type":"print","value":"0911-0119"},{"type":"electronic","value":"1435-5914"}],"subject":[],"published":{"date-parts":[[2025,10,30]]},"assertion":[{"value":"10 February 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 October 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 October 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author has no relevant financial or non-financial interests to disclose.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing Interests"}}],"article-number":"123"}}