Extremal numbers of hypergraph suspensions of even cycles

Sayan Mukherjee

Abstract: For fixed k≥2<math><mrow is="true"><mi is="true">k</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">≥</mo><mn is="true">2</mn></mrow></math>, determining the order of magnitude of the number of edges in an n<math><mi is="true">n</mi></math>-vertex bipartite graph not containing C2k<math><msub is="true"><mrow is="true"><mi is="true">C</mi></mrow><mrow is="true"><mn is="true">2</mn><mi is="true">k</mi></mrow></msub></math>, the cycle of length 2k<math><mrow is="true"><mn is="true">2</mn><mi is="true">k</mi></mrow></math>, is a long-standing open problem. We consider an extension of this problem to triple systems. In particular, we prove that the maximum number of triples in an n<math><mi is="true">n</mi></math>-vertex triple system which does not contain a C6<math><msub is="true"><mrow is="true"><mi is="true">C</mi></mrow><mrow is="true"><mn is="true">6</mn></mrow></msub></math> in the link of any vertex, has order of magnitude n7/3<math><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">7</mn><mo is="true">/</mo><mn is="true">3</mn></mrow></msup></math>. Additionally, we construct new families of dense C6<math><msub is="true"><mrow is="true"><mi is="true">C</mi></mrow><mrow is="true"><mn is="true">6</mn></mrow></msub></math>-free bipartite graphs with n<math><mi is="true">n</mi></math> vertices and n4/3<math><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">4</mn><mo is="true">/</mo><mn is="true">3</mn></mrow></msup></math> edges in order of magnitude.