On an extremal problem for locally sparse multigraphs

A multigraph G is an (s, q)-graph if every s-set of vertices in G supports at most q edges of G, counting multiplicities. Mubayi and Terry posed the problem of determining the maximum of the product of the edge-multiplicities in an (s, q)-graph on n vertices. We give an asymptotic solution to this problem for the family (s, q) = (2r, a ( 2r 2 ) + ex(2r,Kr+1)− 1) with r, a ∈ Z≥2. This asymptotically confirms an infinite family of cases in a conjecture of Day, Treglown and the author.

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