An Improved Error Term for Tur$\acute{\mathrm{a}}$n Number of Expanded Non-degenerate 2-graphs

Abstract

For a 2-graph $F$, let $H_F^{(r)}$ be the $r$-graph obtained from $F$ by enlarging each edge with a new set of $r-2$ vertices. We show that if $\chi (F)=\ell >r\ge 2$, then $\mathrm{ex}(n,H_F^{(r)})=t_r(n,\ell -1)+\Theta (\mathrm{biex}(n,F)n^{r-2}),$ where $t_r(n,\ell -1)$ is the number of edges of an $n$-vertex complete balanced $\ell -1$ partite $r$-graph and $\mathrm{biex}(n,F)$ is the extremal number of the decomposition family of $F$. Since $\mathrm{biex}(n,F)=O(n^{2-\gamma })$ for some $\gamma >0$, this improves on the bound $\mathrm{ex}(n,H_F^{(r)})=t_r(n,\ell -1)+o(n^r)$ by Mubayi (2016). Furthermore, our result implies that $\mathrm{ex}(n,H_F^{(r)})=t_r(n,\ell -1)$ when $F$ is edge-critical, which is an extension of the result of Pikhurko (2013).