On non-degenerate Tur\'an problems for expansions

TL;DR

This paper investigates the maximum number of edges in large r-uniform hypergraphs avoiding certain expanded graphs, establishing bounds and exact values for specific cases using a new structural theorem.

Contribution

It introduces a structure theorem that links Turán numbers of expanded graphs to classical extremal graph problems, enabling exact calculations for large n.

Findings

Established that ex_r(n,F^{(r)+}) equals ex(n,K_r,F) plus a lower order term.

Proved bounds on the maximum edges in F^{(r)+}-free r-graphs.

Determined exact extremal numbers for certain graphs F and large n.

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 $\Omega(n^{r-1})$ (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 $r\chi(F)$ and sufficiently large $n$.