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Counting cliques without generalized theta graphs
The \textit{generalized Tur\'an number} is the maximum
possible number of copies of in an -free graph on vertices for any
two graphs and . For the book graph , there is a close connection
between \ex(n,K_3,B_t) and the Ruzsa-Szemer\'edi triangle removal lemma.
Motivated by this, in this paper, we study the generalized Tur\'an problem for
generalized theta graphs, a natural extension of book graphs. Our main result
provides a complete characterization of the magnitude of \ex(n,K_3,H) when
is a generalized theta graph, indicating when it is quadratic, when it is
nearly quadratic, and when it is subquadratic. Furthermore, as an application,
we obtain the exact value of \ex(n, K_r, kF), where is an edge-critical
generalized theta graph, and , extending several recent results
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