On the jumping constant conjecture for multigraphs

Let G be an infinite family of graphs closed under taking subgraphs. For each n, set = max{|E(G)|: G∈G,|V(G)| = n} nsity τ(G) of a family G be defined by limn→∞ε(n,G)n2 The set of all possible densities of graph families was determined by Erdös and Stone in 1946. The analogous problem for multigraphs with multiplicity of edges not exceeding q (where q = 2, 3, …) appears to be much harder. In 1973 Brown, Erdös, and Simonovits conjectured that the set of all possible densities is well-ordered. They verified their conjecture for q = 2. If q = 1, this follows from the Erdös-Stone theorem. We disprove this conjecture for all q ≥ 4.