On extremal problems on multigraphs

Abstract: An $(n,s,q)$-graph is an $n$-vertex multigraph in which every $s$-set of vertices spans at most $q$ edges. Erd\H{o}s initiated the study of maximum number of edges of $(n,s,q)$-graphs, and the extremal problem on multigraphs has been considered since the 1990s. The problem of determining the maximum product of the edge multiplicities in $(n,s,q)$-graphs was posed by Mubayi and Terry in 2019. Recently, Day, Falgas-Ravry and Treglown settled a conjecture of Mubayi and Terry on the case $(s,q)=(4, 6a + 3)$ of the problem (for $a \ge 2$), and they gave a general lower bound construction for the extremal problem for many pairs $(s, q)$, which they conjectured is asymptotically best possible. Their conjecture was confirmed exactly or asymptotically for some specific cases. In this paper, we consider the case that $(s,q)=(5,\binom{5}{2}a+4)$ and $d=2$ of their conjecture, partially solve an open problem raised by Day, Falgas-Ravry and Treglown. We also show that the conjecture fails for $n=6$, which indicates for the case that $(s,q)=(5,\binom{5}{2}a+4)$ and $d=2$, $n$ needs to be sufficiently large for the conjecture to hold.