The large ambient multiplicity conjecture for the Mubayi–Terry problem

Let $s\in\mathbb{Z}_{\geq 2}$ and let $q$ be an integer with $0\leq q<\binom{s}{2}$. A generalized Turán pattern is a pattern $T=\mathrm{T}[\mathbf{r};a]$ whose blow-ups are $(s,a\binom{s}{2}+q)$-graphs. Large ambient multiplicity conjecture. There exists a unique generalized Turán pattern $T=\mathrm{T}[\mathbf{r};a]$ such that its blow-ups are $(s,a\binom{s}{2}+q)$-graphs and, for all sufficiently large $a$,

$$\operatorname{ex}_\Pi\bigl(n,s,a\binom{s}{2}+q\bigr)=\Pi_n(\mathrm{T}[\mathbf{r};a])^{1+o(1)}.$$

This conjecture seeks a complete description of the Mubayi–Terry problem for sufficiently large ambient edge multiplicity. The preceding results establish substantial special cases, but the asserted uniqueness and asymptotic optimality in full generality remain open.