Abstract:For graphs $F$ and $H$, let $\ex(n,F,H)$ denote the maximum number of copies of $F$ in an $n$-vertex $H$-free graph. Janzer, Longbrake and Yepremyan recently proved that, for fixed $3<a\le b$ and sufficiently large $t$, \[
\ex(n,K_{a,b},K_{3,t})=\Theta(n^3). \] We make their threshold explicit, showing that this conclusion holds for all $t\ge \tau(b):=2\max\{3,\lceil b/2\rceil\}+1.$ In particular, for every even $b\ge 6$, this matches the necessary threshold $t=b+1$. The main new ingredient is an explicit finite-field point set whose plane sections are controlled directly, rather than through a general bounded-complexity algebraic lemma. This direct line-and-conic section analysis gives the required \(K_{3,t}\)-freeness while preserving many coplanar \(b\)-element subsets.
From: Hezhi Wang [view email]
[v1]
Wed, 17 Jun 2026 15:54:48 UTC (16 KB)
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