Counting cliques without generalized theta graphs

Abstract

The \textit{generalized Turán number} $\mathrm{ex}(n, T, F)$ is the maximum possible number of copies of $T$ in an $F$-free graph on $n$ vertices for any two graphs $T$ and $F$. For the book graph $B_t$, there is a close connection between $\ex(n,K_3,B_t)$ and the Ruzsa-Szemerédi triangle removal lemma. Motivated by this, in this paper, we study the generalized Turán 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 $H$ 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 $F$ is an edge-critical generalized theta graph, and $3\le r\le k+1$, extending several recent results.

Figures (6)

• Figure 1: The generalized theta graphs $\Theta(1,2,3,5)$ and $\Theta(1,2,2,3)$.
• Figure 2: The growth process of a path of length $p_i$ derived from $T_i$.
• Figure 3: The illustration of Subcase 1.1.
• Figure 4: The illustration of Subcase 1.2.
• Figure 5: The illustration of Subcase 1.3.

Theorems & Definitions (44)

• Definition 1.1
• Theorem 1.2
• Remark 1
• Remark 2
• Theorem 1.3
• Definition 2.1: $k$-tree, Beineke and Pippert beineke1969number
• Theorem 2.2: Kővári, Sós and Turán kHovari1954problem
• Theorem 2.3: Simonovits simonovits1968method
• Theorem 2.4: Erdős-Simonovits erdos1966limit, Füredi furedi2015proof
• Lemma 3.1