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Statement

If a model for integers $0<a\le b$ exists, then a finite bipartite graph $G$ satisfies

$$\operatorname{ex}(n,G)=\Theta(n^{2-a/b}).$$

The implicit positive lower and upper constants and the eventual threshold may depend on $a,b$ and the chosen graph. The conclusion concerns a single forbidden graph, not a finite forbidden family.

Proof

A model has nonempty internal set and satisfies the required balance condition. [[extremal_graph_theory/adamczewski_2026_erdos571/proposition_2_1|Proposition 2.1]] therefore supplies one integer $t\ge1$ for which

$$\operatorname{ex}(n,F^{(t)})=\Omega(n^{2-a/b}).$$

By the model definition, the matching upper bound holds for every positive $t$, so it holds for this particular $t$. Combining their eventual thresholds gives the asserted two-sided bound with $G=F^{(t)}$. The graph is finite and bipartite by the elementary rooted-power facts. If desired, an arbitrary labeling of its vertices identifies it with a simple graph on $\{0,\ldots,q-1\}$ for some integer $q$; relabeling does not change the extremal number.

Source and dependencies

The preliminary exposition, Proposition 2.2, p. 2. The pinned formal source uses RootedUpperModels.realization, lines 4967–4979, and the relabeling lemma RationalKST.finite_realization, lines 4845–4858. The internal-set nonemptiness missing from the PDF's model definition is included explicitly in the compiled definition, as it is in the formal source.

Bears on. #571.