The geometric stability conjecture for the Mubayi–Terry problem

Let $s\geq 2$ and let $q$ be a non-negative integer. An $(s,q)$-graph is a multigraph satisfying the paper's $(s,q)$-constraint, and $P(G)$ denotes the product of its edge multiplicities. A product-optimal blow-up of a multigraph pattern is a blow-up attaining the asymptotically optimal product density. Geometric stability conjecture. For every pair $(s,q)$, there exists a unique multigraph pattern $P$ such that every $(s,q)$-graph $G$ on $n$ vertices satisfying

$$P(G)\geq \operatorname{ex}_\Pi(s,q)^{\binom{n}{2}+o(n^2)}$$

lies within $o(n^2)$ edit distance of a product-optimal blow-up of $P$. Furthermore, if $a\binom{s}{2}\leq q<(a+1)\binom{s}{2}$, then every edge of $P$ has multiplicity at most $a+1$, while every loop of $P$ has multiplicity at most $a$. The paper ends with this and other open problems; the conjecture is intended as a general stability principle for near-extremal multigraphs.