{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T07:24:31Z","timestamp":1787210671271,"version":"build-2736575974"},"reference-count":6,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T00:00:00Z","timestamp":1781222400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T00:00:00Z","timestamp":1781222400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"HUN-REN Alfr\u00e9d R\u00e9nyi Institute of Mathematics"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Period Math Hung"],"published-print":{"date-parts":[[2026,9]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    If a family\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {F}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>F<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    of\n                    <jats:italic>k<\/jats:italic>\n                    -element subsets of an\n                    <jats:italic>n<\/jats:italic>\n                    -element set is intersecting, then the sum of the sizes of the pairwise intersections of any\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\ell $$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mi>\u2113<\/mml:mi>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    members of the family is at least\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${\\ell \\atopwithdelims ()2}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mfenced>\n                            <mml:mfrac>\n                              <mml:mi>\u2113<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mfrac>\n                          <\/mml:mfenced>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . The classic result of Erd\u0151s, Ko and Rado says that under the condition\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$2k\\le n$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>\u2264<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    an intersecting family of\n                    <jats:italic>k<\/jats:italic>\n                    -element subsets of an\n                    <jats:italic>n<\/jats:italic>\n                    -element set cannot have more than\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${n-1\\atopwithdelims ()k-1}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mfenced>\n                            <mml:mfrac>\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:mrow>\n                                <mml:mi>k<\/mml:mi>\n                                <mml:mo>-<\/mml:mo>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:mfrac>\n                          <\/mml:mfenced>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    members. Is this weaker condition for the sum of the sizes of the pairwise intersections sufficient to have the conclusion of the Erd\u0151s\u2013Ko\u2013Rado theorem? We will see that much more is true, the bound\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${\\ell \\atopwithdelims ()2}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mfenced>\n                            <mml:mfrac>\n                              <mml:mi>\u2113<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:mfrac>\n                          <\/mml:mfenced>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    can be replaced by\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${\\ell -1\\atopwithdelims ()2}+1$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mfenced>\n                              <mml:mfrac>\n                                <mml:mrow>\n                                  <mml:mi>\u2113<\/mml:mi>\n                                  <mml:mo>-<\/mml:mo>\n                                  <mml:mn>1<\/mml:mn>\n                                <\/mml:mrow>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mfrac>\n                            <\/mml:mfenced>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    .\n                  <\/jats:p>","DOI":"10.1007\/s10998-026-00724-x","type":"journal-article","created":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T14:56:05Z","timestamp":1781276165000},"page":"1-5","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Sharpening of the Erd\u0151s\u2013Ko\u2013Rado theorem"],"prefix":"10.1007","volume":"93","author":[{"given":"P\u00e9ter","family":"Frankl","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gyula O. H.","family":"Katona","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kartal","family":"Nagy","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,6,12]]},"reference":[{"key":"724_CR1","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1093\/qmath\/12.1.313","volume":"12","author":"P Erd\u0151s","year":"1961","unstructured":"P. Erd\u0151s, C. Ko, R. Rado, Intersection theorems for systems of finite sets. Q. J. Math. Oxf. II Ser 12, 313\u2013318 (1961)","journal-title":"Q. J. Math. Oxf. II Ser"},{"key":"724_CR2","unstructured":"P. Frankl, The Erd\u0151s\u2013Ko\u2013Rado theorem is true for $$n=ckt$$, in: Combinatorics, Vol I. Proc. Fifth Hungarian Coll. 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