Proves ex(n, K_{a,b}, K_{3,b+1}) = Θ_{a,b}(n^3) for odd b ≥ 5 and 3 < a ≤ b via a projective geometry construction over finite fields.
On the generalized Tur\'an number of complete bipartite graphs
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abstract
For graphs $F$ and $H$, the generalized Tur\'an 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 $\mathrm{ex}(n,K_{a,b},K_{s,t})=\Theta(n^s)$. 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 $\mathrm{ex}(n,F,H)=\Theta(n^r)$ holds for some graph $H$.
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2026 1verdicts
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On the generalized Tur\'{a}n number of the complete bipartite graph $K_{3,b+1}$
Proves ex(n, K_{a,b}, K_{3,b+1}) = Θ_{a,b}(n^3) for odd b ≥ 5 and 3 < a ≤ b via a projective geometry construction over finite fields.