Triangles in graphs without bipartite suspensions.

Given graphs T and H , the generalized Turán number ex (n , T , H) 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 , K 3 , H) 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 H ˆ 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 , K 3 , H ˆ) 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. [ABSTRACT FROM AUTHOR]