Jumps and Nonjumps in Multigraphs
In this paper we consider an extremal problem regarding multigraphs with edge multiplicity bounded by a positive integer $q$. Given a family $\mathscr{F}$ of $q$-multigraphs, define $ex(n,\mathscr{F})$ to be the maximum number of edges (counting multiplicities) that a $q$-multigraph on $n$ vertices can have without containing a copy of any $F \in \mathscr{F}$ (not necessarily induced). It is well known that $\tau(\mathscr{F}) = \lim_{n \to \infty} ex(n,\mathscr{F})/\binom{n}{2}$ exists for every family $\mathscr{F}$ (finite or infinite). Let $\mathscr{T} = \{\tau(\mathscr{F}) : \mathscr{F} \textrm{ is a family of $q$-multigraphs}\}$. We say the number $\alpha$, $0 \leq \alpha \alpha$, then $\alpha' \geq \alpha + c$. The Erdos--Stone theorem implies that for $q=1$, every $\alpha \in [0,1)$ is a jump. The problem of determining the set of jumps for $q \geq 2$ appears to be much ha...