On extremal problems on multigraphs (Q6640956)

An \((n, s, q)\)-graph is an \(n\)-vertex multigraph in which every \(s\)-set of vertices spans at most \(q\) edges. The problem of determining the maximum product of the edge multiplicities in \((n, s, q)\)-graphs was posed by \textit{D. Mubayi} and \textit{C. Terry} [Comb. Probab. Comput. 28, No. 2, 303--324 (2019; Zbl 1434.05078)]. Recently, \textit{A. N. Day} et al. [J. Comb. Theory, Ser. B 154, 1--48 (2022; Zbl 1487.05139)] settled a conjecture of Mubayi and Terry [loc. cit.] on the case \((s, q) = (4, 6a+3)\) of the problem (for \(a \geq 2\)), 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, the authors consider the case that \((s, q) = (5, 10a + 4)\) and \(d = 2\) of their conjecture and partially solve an open problem raised by Day et al. [loc, cit,]. They also show that the conjecture fails for \(n = 6\), which indicates for the case that \((s, q) = (5, 10a + 4)\) and \(d = 2\), \(n\) needs to be sufficiently large for the conjecture to hold.