Triangles in graphs without bipartite suspensions

Dhruv Mubayi, Sayan Mukherjee

Abstract: Given graphs T and H, the generalized Turán number ex(n,T,H)<math><mtext is="true">ex</mtext><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo is="true">,</mo><mi is="true">T</mi><mo is="true">,</mo><mi is="true">H</mi><mo stretchy="false" is="true">)</mo></math> is the maximum number of copies of T in an n-vertex graph with no copies of H. Alon and Shikhelman, using a result of Erdős, determined the asymptotics of ex(n,K3,H)<math><mtext is="true">ex</mtext><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo is="true">,</mo><msub is="true"><mrow is="true"><mi is="true">K</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msub><mo is="true">,</mo><mi is="true">H</mi><mo stretchy="false" is="true">)</mo></math> when the chromatic number of H is greater than three and proved several results when H is bipartite. We consider this problem when H has chromatic number three. Even this special case for the following relatively simple three chromatic graphs appears to be challenging. The suspension Hˆ<math><mover accent="true" is="true"><mrow is="true"><mi is="true">H</mi></mrow><mrow is="true"><mo is="true">ˆ</mo></mrow></mover></math> of a graph H is the graph obtained from H by adding a new vertex adjacent to all vertices of H. We give new upper and lower bounds on ex(n,K3,Hˆ)<math><mtext is="true">ex</mtext><mo stretchy="false" is="true">(</mo><mi is="true">n</mi><mo is="true">,</mo><msub is="true"><mrow is="true"><mi is="true">K</mi></mrow><mrow is="true"><mn is="true">3</mn></mrow></msub><mo is="true">,</mo><mover accent="true" is="true"><mrow is="true"><mi is="true">H</mi></mrow><mrow is="true"><mo is="true">ˆ</mo></mrow></mover><mo stretchy="false" is="true">)</mo></math> when H is a path, even cycle, or complete bipartite graph. One of the main tools we use is the triangle removal lemma, but it is unclear if much stronger statements can be proved without using the removal lemma.