On a hypergraph Tur\'an problem of Balogh-Bohman-Bollob\'as-Zhao

Let $S$ and $T$ be disjoint sets with $\mathrm{\mid }S\mathrm{\mid }=i$ and $\mathrm{\mid }T\mathrm{\mid }=r-1$ for $2\le i\le r-1$, and let $B_i^{(r)}$ be the $r$-graph on $S\cup T$ whose edges are the $r$-subsets containing $S$ or $T$. We study the deficit $q_{r,i}:=1-\pi (B_i^{(r)})$ in its Tur'an density. Balogh, Bohman, Bollob'as, and Zhao previously obtained bounds for these deficits with logarithmic gaps near both ends of the sequence $B_i^{(r)}$, namely, when $i=O(1)$ or $i=r-O(1)$. We close these gaps by showing that, as $r\to \mathrm{\infty }$, for every fixed integer $a\ge 1$, $q_{r,a+1}={\mathrm{\Theta }}_a(r^{-a})$, and for every fixed integer $b\ge 2$, $q_{r,r-b}={\mathrm{\Theta }}_b(r^{-b}\log r)$.