How unproportional must a graph be?

Humberto Naves, Oleg Pikhurko, Alex Scott

Abstract: Let uk(G,p)<math><msub is="true"><mrow is="true"><mi is="true">u</mi></mrow><mrow is="true"><mi is="true">k</mi></mrow></msub><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">,</mo><mi is="true">p</mi><mo is="true">)</mo></mrow></math> be the maximum over all k<math><mi is="true">k</mi></math>-vertex graphs F<math><mi is="true">F</mi></math> of by how much the number of induced copies of F<math><mi is="true">F</mi></math> in G<math><mi is="true">G</mi></math> differs from its expectation in the binomial random graph with the same number of vertices as G<math><mi is="true">G</mi></math> and with edge probability p<math><mi is="true">p</mi></math>. This may be viewed as a measure of how close G<math><mi is="true">G</mi></math> is to being p<math><mi is="true">p</mi></math>-quasirandom. For a positive integer n<math><mi is="true">n</mi></math> and 0<p<1<math><mn is="true">0</mn><mo is="true">&lt;</mo><mi is="true">p</mi><mo is="true">&lt;</mo><mn is="true">1</mn></math>, let D(n,p)<math><mi is="true">D</mi><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">p</mi><mo is="true">)</mo></mrow></math> be the distance from pn2<math><mi is="true">p</mi><mfenced close=")" open="(" is="true"><mrow is="true"><mfrac linethickness="0" is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac></mrow></mfenced></math> to the nearest integer. Our main result is that, for fixed k≥4<math><mi is="true">k</mi><mo is="true">≥</mo><mn is="true">4</mn></math> and for n<math><mi is="true">n</mi></math> large, the minimum of uk(G,p)<math><msub is="true"><mrow is="true"><mi is="true">u</mi></mrow><mrow is="true"><mi is="true">k</mi></mrow></msub><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">,</mo><mi is="true">p</mi><mo is="true">)</mo></mrow></math> over n<math><mi is="true">n</mi></math>-vertex graphs has order of magnitude Θ(max{D(n,p),p(1−p)}nk−2)<math><mi is="true">Θ</mi><mrow is="true"><mo class="bigl" fence="true" is="true">(</mo><mrow is="true"><mo class="qopname" is="true">max</mo><mrow is="true"><mo is="true">{</mo><mi is="true">D</mi><mrow is="true"><mo is="true">(</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">p</mi><mo is="true">)</mo></mrow><mo is="true">,</mo><mi is="true">p</mi><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><mo is="true">−</mo><mi is="true">p</mi><mo is="true">)</mo></mrow><mo is="true">}</mo></mrow><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mi is="true">k</mi><mo is="true">−</mo><mn is="true">2</mn></mrow></msup></mrow><mo class="bigr" fence="true" is="true">)</mo></mrow></math> provided that p(1−p)n1∕2→∞<math><mi is="true">p</mi><mrow is="true"><mo is="true">(</mo><mn is="true">1</mn><mo is="true">−</mo><mi is="true">p</mi><mo is="true">)</mo></mrow><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">1</mn><mo is="true">∕</mo><mn is="true">2</mn></mrow></msup><mo is="true">→</mo><mi is="true">∞</mi></math>.