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 F of q-multigraphs, define ex(n,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 ∈ F (not necessarily induced). It is well known that τ(F ) = limn→∞ ex(n,F )/ (n 2 ) exists for every family F (finite or infinite). Let T = {τ(F ) : F is a family of q-multigraphs}. We say the number α, 0 ≤ α < q is a jump for q if there exists a constant c = c(α, q) such that if α′ ∈ T such that α′ > α then α′ ≥ α+ c. The Erdős-Stone theorem implies that for q = 1, every α ∈ [0, 1) is a jump. The problem of determining the set of jumps for q ≥ 2 appears to be much harder. In a sequence of papers by Erdős, Brown, Simonovits and separately Sidorenko, the authors established that every α is a jump for q = 2 leaving the question whether the same is true for q ≥ 3 unresolved. A later result of Rödl and Sidorenko in [10] gave a negative answer establishing that for q ≥ 4 some values of α are not jumps. The problem of whether or not every α ∈ [0, 3) is a jump for q = 3 has remained open. We give a partial positive result in this paper proving that every α ∈ [0, 2) is a jump for all q ≥ 3. Additionally, we extend the results of [10] by showing that, given any rational number r with 0 < r ≤ 1, that (q − r) is not a jump for any q sufficiently large.