The maximum diameter of pure simplicial complexes and pseudo-manifolds

Francisco Criado, Francisco Santos

Abstract: We construct d-dimensional pure simplicial complexes and pseudo-manifolds (with-out boundary) with n vertices whose combinatorial diameter grows as cdnd1<math><msub is="true"><mrow is="true"><mi is="true">c</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi is="true">d</mi><mn is="true">1</mn></mrow></msup></math> for a constant cd<math><msub is="true"><mrow is="true"><mi is="true">c</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub></math> depending only on d, which is the maximum possible growth. Moreover, the constant cd<math><msub is="true"><mrow is="true"><mi is="true">c</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub></math> is optimal modulo a singly exponential factor in d. The pure simplicial complexes improve on a construction of the second author that achieved cdn2d/3<math><msub is="true"><mrow is="true"><mi is="true">c</mi></mrow><mrow is="true"><mi is="true">d</mi></mrow></msub><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn><mi is="true">d</mi><mo stretchy="false" is="true">/</mo><mn is="true">3</mn></mrow></msup></math>. For pseudo-manifolds without boundary, as far as we know, no construction with diameter greater than n2<math><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup></math> was previously known.