Day–Falgas-Ravry–Treglown multigraph extremal product conjecture

For positive integers $a$ and $r$, let $d$ satisfy $0\le d\le a-1$. For a multigraph $G$, write $P(G)$ for its product of edge multiplicities, and let $ex_\Pi(n,s,q)$ denote the maximum possible value of $P(G)$ over multigraphs on $n$ vertices with no submultigraph on $s$ vertices having edge-multiplicity sum exceeding $q$. Let $\mathcal{T}_{r,d}(a,n)$ be the family of multigraphs whose vertices admit a partition into $r$ parts, with multiplicities $a-d$ within the first part, $a$ within each other part, and $a+1$ between distinct parts; define

$${\Sigma}_{r,d}(a,n):=\max\left\lbrace e(G):G\in\mathcal{T}_{r,d}(a,n)\right\rbrace,$$

and

$$\Pi_{r,d}(a,n):=\max\left\lbrace P(G):G\in\mathcal{T}_{r,d}(a,n)\right\rbrace.$$

Day–Falgas-Ravry–Treglown conjecture. For all integers $a,r,s,d$ with $a,r\geq 1$, $d\in[0,a-1]$, $s\geq(r-1)(d+1)+2$, and all sufficiently large $n$,

$$ex_\Pi(n,s,{\Sigma}_{r,d}(a,s))=\Pi_{r,d}(a,n).$$

This conjecture proposes that, in the stated parameter range, the multigraphs in the construction $\mathcal{T}_{r,d}(a,n)$ maximize the product of edge multiplicities subject to the corresponding local edge-multiplicity constraint. The source gives no resolution, so the conjecture is recorded as open.