On the generalized Turán number of complete bipartite graphs

2026-06-08

AI Summary

This study investigates the asymptotic behavior of the generalized Turán number $\mathrm{ex}(n, K_{a,b}, K_{s,t})$, focusing on the cases where $s = 2$ or $3$ and $t$ is sufficiently large. By employing constructions from extremal graph theory, counting techniques, and asymptotic analysis, the authors establish that $\mathrm{ex}(n, K_{a,b}, K_{s,t}) = \Theta(n^s)$ whenever $s \in \{2,3\}$, $s < a \leq b$, and $t$ is large enough, thereby resolving an open problem posed by Spiro concerning $K_{3,3}$ and $K_{2,t}$. Moreover, they confirm a conjecture on the existence of power-law exponents by showing that for any graph $F$ with at least one edge, there exist infinitely many real numbers $r$ such that $\mathrm{ex}(n,F,H) = \Theta(n^r)$.

Abstract

For graphs $F$ and $H$, the generalized Turán number $\mathrm{ex}(n,F,H)$ denotes the maximum number of copies of $F$ in an $H$-free graph on $n$ vertices. We prove that if $s\in \{2,3\}$, $s< a\leq b$ and $t$ is sufficiently large, then $\mathrm{ex}(n,K_{a,b},K_{s,t})=Θ(n^s)$. The $s=2$, $a=b=3$ case of this result answers a question of Spiro. Proving another conjecture of Spiro, we show that for every graph $F$ with at least one edge, there exist infinitely many real numbers $r$ such that $\mathrm{ex}(n,F,H)=Θ(n^r)$ holds for some graph $H$.

Problem

• generalized Turán number
• complete bipartite graphs
• extremal graph theory
• asymptotic order

Innovation

• generalized Turán number
• complete bipartite graphs
• asymptotic extremal graph theory
• forbidden subgraphs
• graph exponents