An $(n,s,q)$-graph is an n-vertex multigraph in which every s-set of vertices spans at most q edges. Turán-type questions on the maximum of the sum of the edge multiplicities in such multigraphs have been studied since the 1990s. More recently, Mubayi and Terry (2019) [13] posed the problem of determining the maximum of the product of the edge multiplicities in $(n,s,q)$-graphs. We give a general lower bound construction for this problem for many pairs $(s,q)$, which we conjecture is asymptotically best possible. We prove various general cases of our conjecture, and in particular we settle a conjecture of Mubayi and Terry on the $(s,q)=(4,6a+3)$ case of the problem (for $a\ge 2$); this in turn answers a question of Alon. We also determine the asymptotic behaviour of the problem for ‘sparse’ multigraphs (i.e. when $q\le 2\left(\begin{array}{c}s\\ 2\end{array}\right)$). Finally we introduce some tools that are likely to be useful for attacking the problem in general.

一个 $(n,秒,q)$-graph 是一个n -顶点多重图，其中每个s -顶点集最多跨越q 条边。自 1990 年代以来，已经研究了关于此类多重图中边多重性总和的最大值的 Turán 类型问题。最近，Mubayi 和 Terry (2019) [13] 提出了确定边缘多重性乘积的最大值的问题$(n,秒,q)$-图表。我们为这个问题的许多对给出了一个一般的下界构造$(秒,q)$，我们推测这是渐近最好的。我们证明了我们猜想的各种一般情况，特别是我们解决了穆巴伊和特里的猜想$(秒,q)=(4,6\mathrm{一个}+3)$ 问题的情况（对于 $\mathrm{一个}\ge 2$); 这反过来回答了阿隆的一个问题。我们还确定了“稀疏”多重图问题的渐近行为（即，当$q\le 2\left(\begin{array}{c}秒\\ 2\end{array}\right)$）。最后，我们介绍一些可能对解决一般问题有用的工具。