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Statement

Let $F$ be a finite rooted graph with internal set $A\ne\varnothing$ and root set $R$. Fix integers $0<a\le b$ satisfying

$$b|S|\le a e_F(S)\qquad(S\subseteq A).$$

Write $r=|R|$, $e=e(F)$, and choose $d$ with

$$(br+1)e\le d+1.$$

Let $K_0\subseteq K$ be fields and let $P_1,\ldots,P_a$ be polynomials in $2b$ variables of degree at most $d$, whose coefficients on all monomials of degree at most $d$ are algebraically independent over $K_0$. Let $H$ be their polynomial bipartite graph on two copies of $K^b$.

For every fixed placement of the roots and every fixed assignment of sides to the vertices of $F$, the set of injective edge-preserving extensions to $F$ is finite. For each side assignment, these fiber sizes are uniformly bounded over root placements in $H$. Consequently some integer $t\ge1$ satisfies: $H$ has no copy of $F^{(t)}$.

The integer $t$ here may depend on the field and generic coefficient tuple. A separate compactness argument supplies a common obstruction suitable for finite fields.

Proof of finiteness

Suppose one fiber $S$ were infinite. Apply [[extremal_graph_theory/adamczewski_2026_erdos571/polynomial_compactness|compatible independent points]] with $h=br+1$. In an extension of $K$, this gives $h$ compatible placements of $F$, and a selected coordinate of each placement, with the $h$ selected coordinates algebraically independent over $K$.

The finite polynomial conditions for each placement include every edge equation, equality of its root coordinates with the prescribed ones, and injectivity. For two vertices assigned to the same side, injectivity is the clause that at least one of their $b$ coordinate differences is nonzero; vertices on opposite sides are already distinct. If an edge is assigned within one side, its equation can be written $1=0$, so that side assignment simply has an empty fiber. Compatibility preserves all these conditions. Thus the new placements are injective copies with a common root map.

Let $I$ be the union of their internal images, $E$ the union of their edge images, and $U$ the union of all their vertices. Then

$$|U|\le |I|+r,\qquad |E|\le he\le d+1, \qquad b|I|\le a|E|.$$

The last inequality is [[extremal_graph_theory/adamczewski_2026_erdos571/rooted_union_balance|union balance]]. Let $M$ denote the number of allowed monomials. The $aM$ coefficients together with the $h$ selected coordinates are algebraically independent over $K_0$: the former are independent over $K_0$, and the latter are independent over the larger field $K$ containing them. All selected coordinates belong to vertices in $U$. The [[extremal_graph_theory/adamczewski_2026_erdos571/polynomial_interpolation|coefficient bound]] gives

$$h+a|E|\le b|U|\le b|I|+br\le a|E|+br.$$

Hence $h\le br$, contradicting $h=br+1$. This proves finiteness. The argument remains valid after any extension of $K$, since field embeddings preserve algebraic independence of the coefficients over $K_0$.

Uniformity and exclusion of a power

For a fixed side assignment, the edge and injectivity conditions above are a finite polynomial condition, with root coordinates as parameters and internal coordinates as fiber variables. The fibers remain finite in every extension field, as just proved. The [[extremal_graph_theory/adamczewski_2026_erdos571/polynomial_compactness|uniform fiber bound]] supplies a number $N_\kappa$ for each side assignment $\kappa:V(F)\to\{0,1\}$.

Set $t=1+\sum_\kappa N_\kappa$. A copy of $F^{(t)}$ in $H$ would give $t$ rooted copies of $F$ with the same root coordinates. Count these according to their side assignment. In class $\kappa$ there are at most $N_\kappa$ possibilities. They are genuinely distinct placements: the images of any fixed internal vertex in the different layers are distinct, using $A\ne\varnothing$ and injectivity of the whole power. This contradicts the choice of $t$.

Source and dependencies

This is the finite-fiber core of Proposition 2.1, p. 2 of the exposition, expanded from the pinned formal declarations GenericRootedFiber.finite_fiber, lines 2941–3043; GenericUniformFiber.uniform_bound, lines 3273–3330; and GenericPowerFree.exists_power_free, lines 3445–3491. The necessary polynomial clauses are defined in PolynomialCopyConstraints, lines 3145–3249. The preceding linked lemmas give every nonroutine deduction. No estimate on the number of rational points of a variety is assumed.

Bears on. #571.