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Abstract:For graphs $H$ and $F$, the generalized Turán number $\operatorname{ex}(n,H,F)$ denotes the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. We prove a general lifting principle for the case $H=K_3$ and the forbidden graph is a vertex-disjoint union of several copies of a fixed graph. The key hypothesis is a local neighborhood-forcing condition: there is a fixed graph $R$ with $\operatorname{ex}(n,R)=o(n^2)$ such that $F\subseteq K_1\nabla R$. Under this condition, the corresponding single-forbidden-graph asymptotics, together with a construction attaining the relevant extremal triangle and edge densities simultaneously, lift to an asymptotic formula for \(\operatorname{ex}(n,K_3,(s+1)F)\) for every fixed \(s\). We also prove an exact version in terms of a weighted extremal function $\psi_q(m;F)=\max\{t(Q)+q e(Q): |V(Q)|=m,\ Q\text{ is }F\text{-free}\}$. As applications, we recover the exact theorem of Hou, Yang, and Zeng for $(s+1)C_{2k+1}$, derive exact values and characterize the corresponding extremal graphs for \((s+1)\widehat P_4\) and \((s+1)\widehat P_5\), and obtain the asymptotic value for $(s+1)\widehat P_6$. Here \(\widehat P_k\) denotes the suspension of \(P_k\), obtained from \(P_k\) by adding a new vertex adjacent to every vertex of \(P_k\).

Submission history

From: Junjie Wang [view email]
[v1] Wed, 24 Jun 2026 13:03:53 UTC (9 KB)