Counting multiple graphs in generalized Turán problems
Dániel Gerbner
TL;DR
We study counting sums of copies of several subgraphs in $F$-free graphs by defining $ex(n,(H_1,\dots,H_k),F)=\max_G \sum_{i=1}^k \mathcal{N}(H_i,G)$ and its colored variant $ex^{\mathrm{col}}(n,(H_1,\dots,H_k),F)=\max_G \sum_{i=1}^k \mathcal{N}(H_i,G_i)$. Our approach relies on stability results and color-critical-edge analysis to show that near-extremal configurations resemble Turán-type structures, often reducing to complete multipartite extremals via Zykov-type symmetrization. Key findings include color-resistance for all-tuples of cliques in the colored setting, alongside instances where the uncolored analogue fails to align with the maximum of individual $H_i$-extremals. The work connects generalized Turán problems to Berge-$F$ hypergraphs, yielding bounds on ex_r(n, Berge-$F$) in terms of classical Turán numbers and illustrating the utility of stability methods for deriving asymptotics and exact results.
Abstract
We are given graphs $H_1,\dots,H_k$ and $F$. Consider an $F$-free graph $G$ on $n$ vertices. What is the largest sum of the number of copies of $H_i$? The case $k=1$ has attracted a lot of attention. We also consider a colored variant, where the edges of $G$ are colored with $k$ colors. What is the largest sum of the number of copies of $H_i$ in color $i$? Our motivation to study this colored variant is a recent result stating that the Turán number of the $r$-uniform Berge-$F$ hypergraphs is at most the quantity defined above for $k=2$, $H_1=K_r$ and $H_2=K_2$. In addition to studying these new questions, we obtain new results for generalized Turán problems and also for Berge hypergraphs.