On non-degenerate Turán problems for expansions
Dániel Gerbner
TL;DR
This work analyzes non-degenerate Turán problems for the r-uniform expansion F^{(r)+}, proving that for χ(F)=k+1>r the Turán number satisfies $\mathrm{ex}_r(n,F^{(r)+})=\mathrm{ex}(n,K_r,F)+O(n^{r-1})$, with a generalization to $\mathrm{ex}_p(n,K_r^p,F^{(p)+})+O(n^r)$. A structural framework based on shadows, heavy/fat sets, and the ∂^*(t) operator connects the r-graph problem to lower-order cases, enabling exact or near-exact determinations for certain F and large n. The authors also discuss stability connections (to Ma and Qiu; Tan, Xu, and Yan), provide corrected bounds in the Θ(1) biex(F) regime, and present explicit extremal constructions and exact results for special graphs such as B_{k+1,1}. These results bridge classical graph Turán theory with hypergraph expansions and inform generalized Turán problems by relating higher-order extremal quantities to their graph counterparts.
Abstract
The $r$-uniform expansion $F^{(r)+}$ of a graph $F$ is obtained by enlarging each edge with $r-2$ new vertices such that altogether we use $(r-2)|E(F)|$ new vertices. Two simple lower bounds on the largest number $\mathrm{ex}_r(n,F^{(r)+})$ of $r$-edges in $F^{(r)+}$-free $r$-graphs are $Ω(n^{r-1})$ (in the case $F$ is not a star) and $\mathrm{ex}(n,K_r,F)$, which is the largest number of $r$-cliques in $n$-vertex $F$-free graphs. We prove that $\mathrm{ex}_r(n,F^{(r)+})=\mathrm{ex}(n,K_r,F)+O(n^{r-1})$. The proof comes with a structure theorem that we use to determine $\ex_r(n,F^{(r)+})$ exactly for some graphs $F$, every $rχ(F)$ and sufficiently large $n$.