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

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 $\chi (F)=\mathrm{\ell }>r\ge 2$, then $ {\rm ex}(n,H_F^{®})= t_r (n,\ell-1)+ \Theta( {\rm biex}(n,F)n^{r-2}),$where$t_r (n,\ell-1)$isthenumberofedgesofan$n$-vertexcompletebalanced$\ell-1$partite$r$-graphand${\rm biex}(n,F)$istheextremalnumberofthedecompositionfamilyof$F$\mathrm{.}Since${\rm biex}(n,F)=O(n^{2-\gamma})$forsome$\gamma>0$,thisimprovesonthebound${\rm ex}(n,H_F^{®})= t_r (n,\ell-1)+ o(n^r)$ by Mubayi (2016). Furthermore, our result implies that $\mathrm{e}\mathrm{x}(n,H_F^{(r)})=t_r(n,\mathrm{\ell }-1)$ when $F$ is edge-critical, which is an extension of the result of Pikhurko (2013).