{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,6]],"date-time":"2025-05-06T09:32:58Z","timestamp":1746523978798},"reference-count":17,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,5,26]],"date-time":"2023-05-26T00:00:00Z","timestamp":1685059200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,5,26]],"date-time":"2023-05-26T00:00:00Z","timestamp":1685059200000},"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":[[2023,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>For positive integers <jats:italic>s<\/jats:italic>,\u00a0<jats:italic>t<\/jats:italic>,\u00a0<jats:italic>r<\/jats:italic>, let <jats:inline-formula><jats:alternatives><jats:tex-math>$$K_{s,t}^{(r)}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msubsup>\n                    <mml:mi>K<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>s<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>t<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>r<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:msubsup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> denote the <jats:italic>r<\/jats:italic>-uniform hypergraph whose vertex set is the union of pairwise disjoint sets <jats:inline-formula><jats:alternatives><jats:tex-math>$$X,Y_1,\\dots ,Y_t$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>Y<\/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>Y<\/mml:mi>\n                      <mml:mi>t<\/mml:mi>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$|X| = s$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>|<\/mml:mo>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mo>|<\/mml:mo>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mi>s<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$|Y_1| = \\dots = |Y_t| = r-1$$<\/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:mrow>\n                    <mml:msub>\n                      <mml:mi>Y<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                      <mml:mo>=<\/mml:mo>\n                      <mml:mo>\u22ef<\/mml:mo>\n                      <mml:mo>=<\/mml:mo>\n                      <mml:mo>|<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:msub>\n                      <mml:mi>Y<\/mml:mi>\n                      <mml:mi>t<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                      <mml:mo>=<\/mml:mo>\n                      <mml:mi>r<\/mml:mi>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and whose edge set is <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\{\\{x\\} \\cup Y_i: x \\in X, 1\\le i\\le t\\}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>{<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>{<\/mml:mo>\n                      <mml:mi>x<\/mml:mi>\n                      <mml:mo>}<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u222a<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>Y<\/mml:mi>\n                      <mml:mi>i<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>:<\/mml:mo>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mi>X<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mi>i<\/mml:mi>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>}<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The study of the Tur\u00e1n function of <jats:inline-formula><jats:alternatives><jats:tex-math>$$K_{s,t}^{(r)}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msubsup>\n                    <mml:mi>K<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>s<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>t<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>r<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:msubsup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> received considerable interest in recent years. Our main results are as follows. First, we show that <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} \\textrm{ex}\\left( n,K_{s,t}^{(r)}\\right) = O_{s,r}\\left( t^{\\frac{1}{s-1}}n^{r - \\frac{1}{s-1}}\\right) \\end{aligned}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtable>\n                      <mml:mtr>\n                        <mml:mtd>\n                          <mml:mrow>\n                            <mml:mtext>ex<\/mml:mtext>\n                            <mml:mfenced>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:msubsup>\n                                <mml:mi>K<\/mml:mi>\n                                <mml:mrow>\n                                  <mml:mi>s<\/mml:mi>\n                                  <mml:mo>,<\/mml:mo>\n                                  <mml:mi>t<\/mml:mi>\n                                <\/mml:mrow>\n                                <mml:mrow>\n                                  <mml:mo>(<\/mml:mo>\n                                  <mml:mi>r<\/mml:mi>\n                                  <mml:mo>)<\/mml:mo>\n                                <\/mml:mrow>\n                              <\/mml:msubsup>\n                            <\/mml:mfenced>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>O<\/mml:mi>\n                              <mml:mrow>\n                                <mml:mi>s<\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>r<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mfenced>\n                              <mml:msup>\n                                <mml:mi>t<\/mml:mi>\n                                <mml:mfrac>\n                                  <mml:mn>1<\/mml:mn>\n                                  <mml:mrow>\n                                    <mml:mi>s<\/mml:mi>\n                                    <mml:mo>-<\/mml:mo>\n                                    <mml:mn>1<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:mfrac>\n                              <\/mml:msup>\n                              <mml:msup>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mrow>\n                                  <mml:mi>r<\/mml:mi>\n                                  <mml:mo>-<\/mml:mo>\n                                  <mml:mfrac>\n                                    <mml:mn>1<\/mml:mn>\n                                    <mml:mrow>\n                                      <mml:mi>s<\/mml:mi>\n                                      <mml:mo>-<\/mml:mo>\n                                      <mml:mn>1<\/mml:mn>\n                                    <\/mml:mrow>\n                                  <\/mml:mfrac>\n                                <\/mml:mrow>\n                              <\/mml:msup>\n                            <\/mml:mfenced>\n                          <\/mml:mrow>\n                        <\/mml:mtd>\n                      <\/mml:mtr>\n                    <\/mml:mtable>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:disp-formula>for all <jats:inline-formula><jats:alternatives><jats:tex-math>$$s,t\\ge 2$$<\/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>t<\/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> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$r\\ge 3$$<\/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>3<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, improving the power of <jats:italic>n<\/jats:italic> in the previously best bound and resolving a question of Mubayi and Verstra\u00ebte about the dependence of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{ex}(n,K_{2,t}^{(3)})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtext>ex<\/mml:mtext>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msubsup>\n                      <mml:mi>K<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mn>2<\/mml:mn>\n                        <mml:mo>,<\/mml:mo>\n                        <mml:mi>t<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mn>3<\/mml:mn>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:msubsup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> on <jats:italic>t<\/jats:italic>. Second, we show that (1) is tight when <jats:italic>r<\/jats:italic> is even and <jats:inline-formula><jats:alternatives><jats:tex-math>$$t \\gg s$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>\u226b<\/mml:mo>\n                    <mml:mi>s<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. This disproves a conjecture of Xu, Zhang and Ge. Third, we show that (1) is <jats:italic>not<\/jats:italic> tight for <jats:inline-formula><jats:alternatives><jats:tex-math>$$r = 3$$<\/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>=<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, namely that <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{ex}(n,K_{s,t}^{(3)}) = O_{s,t}(n^{3 - \\frac{1}{s-1} - \\varepsilon _s})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtext>ex<\/mml:mtext>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msubsup>\n                        <mml:mi>K<\/mml:mi>\n                        <mml:mrow>\n                          <mml:mi>s<\/mml:mi>\n                          <mml:mo>,<\/mml:mo>\n                          <mml:mi>t<\/mml:mi>\n                        <\/mml:mrow>\n                        <mml:mrow>\n                          <mml:mo>(<\/mml:mo>\n                          <mml:mn>3<\/mml:mn>\n                          <mml:mo>)<\/mml:mo>\n                        <\/mml:mrow>\n                      <\/mml:msubsup>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>O<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mi>s<\/mml:mi>\n                        <mml:mo>,<\/mml:mo>\n                        <mml:mi>t<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msup>\n                        <mml:mi>n<\/mml:mi>\n                        <mml:mrow>\n                          <mml:mn>3<\/mml:mn>\n                          <mml:mo>-<\/mml:mo>\n                          <mml:mfrac>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mrow>\n                              <mml:mi>s<\/mml:mi>\n                              <mml:mo>-<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:mrow>\n                          <\/mml:mfrac>\n                          <mml:mo>-<\/mml:mo>\n                          <mml:msub>\n                            <mml:mi>\u03b5<\/mml:mi>\n                            <mml:mi>s<\/mml:mi>\n                          <\/mml:msub>\n                        <\/mml:mrow>\n                      <\/mml:msup>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (for all <jats:inline-formula><jats:alternatives><jats:tex-math>$$s\\ge 3$$<\/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>\u2265<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>). This indicates that the behaviour of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{ex}(n,K_{s,t}^{(r)})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtext>ex<\/mml:mtext>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msubsup>\n                      <mml:mi>K<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mi>s<\/mml:mi>\n                        <mml:mo>,<\/mml:mo>\n                        <mml:mi>t<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mi>r<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:msubsup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> might depend on the parity of <jats:italic>r<\/jats:italic>. Lastly, we prove a conjecture of Ergemlidze, Jiang and Methuku on the hypergraph analogue of the bipartite Tur\u00e1n problem for graphs with bounded degrees on one side. Our tools include a novel twist on the dependent random choice method as well as a variant of the celebrated norm graphs constructed by Koll\u00e1r, R\u00f3nyai and Szab\u00f3.<\/jats:p>","DOI":"10.1007\/s00493-023-00019-6","type":"journal-article","created":{"date-parts":[[2023,5,26]],"date-time":"2023-05-26T09:04:05Z","timestamp":1685091845000},"page":"429-446","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Asymptotics of the Hypergraph Bipartite Tur\u00e1n Problem"],"prefix":"10.1007","volume":"43","author":[{"given":"Domagoj","family":"Brada\u010d","sequence":"first","affiliation":[]},{"given":"Lior","family":"Gishboliner","sequence":"additional","affiliation":[]},{"given":"Oliver","family":"Janzer","sequence":"additional","affiliation":[]},{"given":"Benny","family":"Sudakov","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,5,26]]},"reference":[{"issue":"2","key":"19_CR1","doi-asserted-by":"publisher","first-page":"280","DOI":"10.1006\/jctb.1999.1906","volume":"76","author":"N Alon","year":"1999","unstructured":"Alon, N., R\u00f3nyai, L., Szab\u00f3, T.: Norm-graphs: variations and applications. 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