Induced subgraph density. II. Sparse and dense sets in cographs

Jacob Fox, Tung Nguyen, Alex D. Scott, Paul D. Seymour

Abstract: A well-known theorem of Rödl says that for every graph H<math><mi is="true">H</mi></math>, and every ɛɛ>0<math><mrow is="true"><mi is="true">ɛ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mn is="true">0</mn></mrow></math>, there exists δ>0<math><mrow is="true"><mi is="true">δ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mn is="true">0</mn></mrow></math> such that if G<math><mi is="true">G</mi></math> does not contain an induced copy of H<math><mi is="true">H</mi></math>, then there exists X⊆V(G)<math><mrow is="true"><mi is="true">X</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">⊆</mo><mi is="true">V</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></mrow></math> with |X|≥δ|G|<math><mrow is="true"><mrow is="true"><mo is="true">|</mo><mi is="true">X</mi><mo is="true">|</mo></mrow><mo linebreak="goodbreak" linebreakstyle="after" is="true">≥</mo><mi is="true">δ</mi><mrow is="true"><mo is="true">|</mo><mi is="true">G</mi><mo is="true">|</mo></mrow></mrow></math> such that one of G[X],G¯[X]<math><mrow is="true"><mi is="true">G</mi><mrow is="true"><mo is="true">[</mo><mi is="true">X</mi><mo is="true">]</mo></mrow><mo is="true">,</mo><mover accent="false" class="mml-overline" is="true"><mrow is="true"><mi is="true">G</mi></mrow><mo accent="true" is="true">¯</mo></mover><mrow is="true"><mo is="true">[</mo><mi is="true">X</mi><mo is="true">]</mo></mrow></mrow></math> has edge-density at most ɛɛ<math><mi is="true">ɛ</mi></math>. But how does δ<math><mi is="true">δ</mi></math> depend on ϵ<math><mi is="true">ϵ</mi></math>? Fox and Sudakov conjectured that the dependence is at most polynomial: that for all H<math><mi is="true">H</mi></math> there exists c>0<math><mrow is="true"><mi is="true">c</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mn is="true">0</mn></mrow></math> such that for all ɛɛ<math><mi is="true">ɛ</mi></math> with ɛ0<ɛ≤1/2<math><mrow is="true"><mn is="true">0</mn><mo linebreak="goodbreak" linebreakstyle="after" is="true">&lt;</mo><mi is="true">ɛ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">≤</mo><mn is="true">1</mn><mo is="true">/</mo><mn is="true">2</mn></mrow></math>, Rödl’s theorem holds with ɛδ=ɛc<math><mrow is="true"><mi is="true">δ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><msup is="true"><mrow is="true"><mi is="true">ɛ</mi></mrow><mrow is="true"><mi is="true">c</mi></mrow></msup></mrow></math>. This conjecture implies the Erdős–Hajnal conjecture, and until now it had not been verified for any non-trivial graphs H<math><mi is="true">H</mi></math>. Our first result shows that it is true when H=P4<math><mrow is="true"><mi is="true">H</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msub></mrow></math>. Indeed, in that case we can take ɛδ=ɛ<math><mrow is="true"><mi is="true">δ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mi is="true">ɛ</mi></mrow></math>, and insist that one of G[X],G¯[X]<math><mrow is="true"><mi is="true">G</mi><mrow is="true"><mo is="true">[</mo><mi is="true">X</mi><mo is="true">]</mo></mrow><mo is="true">,</mo><mover accent="false" class="mml-overline" is="true"><mrow is="true"><mi is="true">G</mi></mrow><mo accent="true" is="true">¯</mo></mover><mrow is="true"><mo is="true">[</mo><mi is="true">X</mi><mo is="true">]</mo></mrow></mrow></math> has maximum degree at most ɛɛ2|G|<math><mrow is="true"><msup is="true"><mrow is="true"><mi is="true">ɛ</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msup><mrow is="true"><mo is="true">|</mo><mi is="true">G</mi><mo is="true">|</mo></mrow></mrow></math>).Second, we will show that every graph H<math><mi is="true">H</mi></math> that can be obtained by substitution from copies of P4<math><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msub></math> satisfies the Fox–Sudakov conjecture. To prove this, we need to work with a stronger property. Let us say H<math><mi is="true">H</mi></math> is viral if there exists c>0<math><mrow is="true"><mi is="true">c</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&gt;</mo><mn is="true">0</mn></mrow></math> such that for all ɛɛ<math><mi is="true">ɛ</mi></math> with ɛ0<ɛ≤1/2<math><mrow is="true"><mn is="true">0</mn><mo linebreak="goodbreak" linebreakstyle="after" is="true">&lt;</mo><mi is="true">ɛ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">≤</mo><mn is="true">1</mn><mo is="true">/</mo><mn is="true">2</mn></mrow></math>, if G<math><mi is="true">G</mi></math> contains at most ɛɛc|G||H|<math><mrow is="true"><msup is="true"><mrow is="true"><mi is="true">ɛ</mi></mrow><mrow is="true"><mi is="true">c</mi></mrow></msup><msup is="true"><mrow is="true"><mrow is="true"><mo is="true">|</mo><mi is="true">G</mi><mo is="true">|</mo></mrow></mrow><mrow is="true"><mrow is="true"><mo is="true">|</mo><mi is="true">H</mi><mo is="true">|</mo></mrow></mrow></msup></mrow></math> copies of H<math><mi is="true">H</mi></math> as induced subgraphs, then there exists X⊆V(G)<math><mrow is="true"><mi is="true">X</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">⊆</mo><mi is="true">V</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></mrow></math> with ɛ|X|≥ɛc|G|<math><mrow is="true"><mrow is="true"><mo is="true">|</mo><mi is="true">X</mi><mo is="true">|</mo></mrow><mo linebreak="goodbreak" linebreakstyle="after" is="true">≥</mo><msup is="true"><mrow is="true"><mi is="true">ɛ</mi></mrow><mrow is="true"><mi is="true">c</mi></mrow></msup><mrow is="true"><mo is="true">|</mo><mi is="true">G</mi><mo is="true">|</mo></mrow></mrow></math> such that one of G[X],G¯[X]<math><mrow is="true"><mi is="true">G</mi><mrow is="true"><mo is="true">[</mo><mi is="true">X</mi><mo is="true">]</mo></mrow><mo is="true">,</mo><mover accent="false" class="mml-overline" is="true"><mrow is="true"><mi is="true">G</mi></mrow><mo accent="true" is="true">¯</mo></mover><mrow is="true"><mo is="true">[</mo><mi is="true">X</mi><mo is="true">]</mo></mrow></mrow></math> has edge-density at most ɛɛ<math><mi is="true">ɛ</mi></math>. We will show that P4<math><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msub></math> is viral, using a “polynomial P4<math><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msub></math>-removal lemma” of Alon and Fox. We will also show that the class of viral graphs is closed under vertex-substitution.Finally, we give a different strengthening of Rödl’s theorem: we show that if G<math><mi is="true">G</mi></math> does not contain an induced copy of P4<math><msub is="true"><mrow is="true"><mi is="true">P</mi></mrow><mrow is="true"><mn is="true">4</mn></mrow></msub></math>, then its vertices can be partitioned into at most ɛ480ɛ−4<math><mrow is="true"><mn is="true">480</mn><msup is="true"><mrow is="true"><mi is="true">ɛ</mi></mrow><mrow is="true"><mo is="true">−</mo><mn is="true">4</mn></mrow></msup></mrow></math> subsets X<math><mi is="true">X</mi></math> such that one of G[X],G¯[X]<math><mrow is="true"><mi is="true">G</mi><mrow is="true"><mo is="true">[</mo><mi is="true">X</mi><mo is="true">]</mo></mrow><mo is="true">,</mo><mover accent="false" class="mml-overline" is="true"><mrow is="true"><mi is="true">G</mi></mrow><mo accent="true" is="true">¯</mo></mover><mrow is="true"><mo is="true">[</mo><mi is="true">X</mi><mo is="true">]</mo></mrow></mrow></math> has maximum degree at most ɛɛ|X|<math><mrow is="true"><mi is="true">ɛ</mi><mrow is="true"><mo is="true">|</mo><mi is="true">X</mi><mo is="true">|</mo></mrow></mrow></math>.