A generalised transference principle

Abstract: The last two decades have witnessed a growing trend towards proving sparse random analogues of combinatorial theorems. One unified approach to proving such theorems, formalised by Conlon and Gowers [Ann. of Math. 2016], involves establishing a 'transference principle' which allows one to translate between robust properties in the dense setting and the sparse $p$-random setting, provided $p$ is not too small. Our results provide a more general transference theorem, extending the results of Conlon and Gowers and also those of Schacht [Ann. of Math. 2016]. Among a variety of other applications, we use this to obtain a sparse counting lemma for graphs and hypergraphs which are not necessarily strictly balanced. Our method achieves asymptotically optimal bounds on the probability $p$, and the probability of success.

• The paper establishes a transference principle for all graphs and hypergraphs under container-style codegree conditions, producing dense models that preserve edge densities and substructure counts at the optimal threshold p ≥ C N^{-1/m₂(H)}.
• The framework uses structure functions, subcounts, compatible colourings, and multiple simultaneous structures to transfer regularity, localised counting, Ramsey multiplicities, and Szemerédi-type results from dense to sparse settings.
• The method removes the strict-balancing restriction, achieves asymptotically optimal failure probabilities, and supports applications including sparse pentagon, Ramsey multiplicity, arithmetic progression, and hypergraph counting theorems.

Context and motivation

Sparse random analogues of classical extremal theorems—Turan, Ramsey, and Szemerédi—have been a central concern of combinatorics since the 1990s. The transference principle, introduced by Green and Tao in their work on arithmetic progressions in the primes and generalised by Conlon and Gowers, provides a unified route to such results: given a subgraph $G$ of a random graph $\Gamma = G(N,p)$, one constructs a dense model $G' \subseteq K_N$ whose relative edge density and $H$-copy counts match those of $G$, so that dense extremal results transfer to the sparse setting. A limitation of the Conlon–Gowers framework is that it handles only strictly balanced graphs $H$ (where the maximum $2$-density $m_2(H)$ is attained uniquely by $H$ itself). Schacht's independent approach removed this restriction but did not yield a transference principle, and the hypergraph container method of Balogh–Morris–Samotij and Saxton–Thomason gave a third route to Turán-type results without counting strength. The paper under review unifies and extends all three lines: it proves an abstract transference principle valid under the same codegree conditions as the container theorems, with asymptotically optimal bounds on both the probability $p$ and the failure probability, and derives from it a full sparse counting lemma for graphs and hypergraphs that need not be strictly balanced (2608.17982).

The main theorem for graphs

For a fixed graph $\Gamma = G(N,p)$0 with $\Gamma = G(N,p)$1, let $\Gamma = G(N,p)$2 denote the probability that $\Gamma = G(N,p)$3 contains more than $\Gamma = G(N,p)$4 copies of $\Gamma = G(N,p)$5. The simplified transference theorem states that there is a constant $\Gamma = G(N,p)$6 such that for $\Gamma = G(N,p)$7, with probability at least $\Gamma = G(N,p)$8, every subgraph $\Gamma = G(N,p)$9 admits a dense model $G' \subseteq K_N$0 on the same vertex set satisfying

$G' \subseteq K_N$1

Two claims of optimality are made, both substantiated. First, the lower bound on $G' \subseteq K_N$2 is asymptotically optimal: for $G' \subseteq K_N$3, all $G' \subseteq K_N$4-copies can be removed from $G' \subseteq K_N$5 by deleting a vanishing fraction of its edges, so no dense model can exist. Second, the failure probability is optimal up to constants: for any non-forest $G' \subseteq K_N$6, $G' \subseteq K_N$7, and no transference statement can succeed with probability better than $G' \subseteq K_N$8, since if $G' \subseteq K_N$9 contains too many $H$0-copies, even $H$1 itself cannot serve as a model. The restriction $H$2 merely excludes matchings, which are handled at the cost of a polylogarithmic loss in $H$3.

The theorem strictly generalises the corresponding result of Conlon and Gowers in two respects: it applies to all graphs $H$4, not merely strictly balanced ones, and it achieves the optimal failure probability. Relative to Schacht and the container method, it provides two-sided counting with a $H$5-factor rather than only existence of copies.

Extensions: structure functions, subcounts, and colours

The full framework operates on $H$6-uniform ordered hypergraphs $H$7 on $H$8; for instance, counting $H$9-copies in $G$0 corresponds to an ordered $G$1-uniform hypergraph on $G$2 nodes. Two additional features are introduced to obtain the sparse counting lemma.

Structure functions are functions $G$3 on the nodes; requiring the dense model to have correct $G$4-weighted counts for a suitable family $G$5 transfers structural information such as $G$6-regularity of pairs. Subcounts are functions $G$7 on the edges; requiring correct $G$8-weighted counts for a family $G$9 permits counting copies in specific places only (e.g. transversal $H$0s through four designated clusters, excluding copies lying within a single pair). Both families may be exponentially large—the framework accommodates $H$1—and the paper notes that exponentially many of each are genuinely needed for the counting lemma. The transference also handles $H$2-colours simultaneously, producing compatible dense models (an $H$3-partition of $H$4), which is essential for Ramsey applications, and extends to multiple structures counted at once.

The general statement (Theorem 1.7 of the paper) requires the codegree conditions $H$5 for each $H$6—the same conditions governing hypergraph containers—and yields, with probability at least $H$7, an $H$8-good dense model for every partition of $H$9. A companion "lower dense model" version drops the upper bound on counts and removes the $2$0 term from the failure probability entirely, giving $2$1. The main technical theorem achieves both the full two-sided statement and the optimal failure probability at the cost of an $2$2-deletion of the random set.

Proof architecture

The proof is a substantial refinement of the Conlon–Gowers approach, conducted in functional language. Sparse sets become scaled indicator functions $2$3 and $2$4; dense models are $2$5-valued functions $2$6 summing to $2$7; and the model quality is measured in a norm $2$8, where $2$9 is the convex hull of "test functions"—structure functions and convolutions $m_2(H)$0 of $m_2(H)$1-bounded functions and their negatives. The paper observes that Conlon and Gowers were "tempted" to use essentially this norm but found it led to difficulties, forcing them to a more complicated capped-convolution norm; resolving those difficulties is one of the paper's main contributions.

The argument proceeds through a chain of reductions:

1. Rounding. A randomised rounding converts $m_2(H)$2-valued dense models to $m_2(H)$3-valued ones, analysed via Bernstein's inequality over the vertices of a split polytope, with large entries controlled by moment bounds.
2. Separating hyperplane. The existence of $m_2(H)$4-valued models is reduced, via the separating hyperplane theorem applied to an appropriate convex set, to an anti-correlation statement: $m_2(H)$5 for all test functions $m_2(H)$6. A rescaling subtlety is handled using bounds on $m_2(H)$7 and $m_2(H)$8.
3. Linearisation. The nonlinearity of $m_2(H)$9 is removed by Stone–Weierstrass polynomial approximation on $H$0, reducing the problem to linear optimisation over a product polytope $H$1; the contribution of entries outside the approximation domain is killed by moment bounds $H$2.
4. Random splitting. Since $H$3 has far too many vertices (already $H$4 for the $H$5-extreme functions) and its vertices depend on the random set, $H$6 and $H$7 are split into $H$8 and $H$9 randomly chosen parts respectively. The resulting split polytope $p$0 contains $p$1 but has far fewer vertices, each depending on only $p$2-many parts of the split; Bernstein's inequality plus a union bound then gives anti-correlation, with the revealed part of each vertex and the $p$3–$p$4 discrepancy handled by moment bounds.
5. Moment bounds. These are established via the Kim–Vu polynomial concentration inequality applied to "book" counts, together with a coupling argument to handle the dependence among split parts. The precise codegree bound is used in exactly one place: bounding the expected book count so that the dominant contribution comes from books whose pages meet only in the spine.
6. Deletion. Exponentially small failure probability for the moment bounds is obtained from the high-probability statement via the Harris inequality, following a deletion trick of Spöhel, Steger and Warnke in the spirit of Rödl and Ruciński.

Applications

The framework yields several results, of which the following are representative.

Sparse random $p$5-density. For any $p$6-uniform hypergraph $p$7 with at least two edges and $p$8, any $p$9-vertex subgraph of $\Gamma = G(N,p)$00 with $\Gamma = G(N,p)$01 edges contains between $\Gamma = G(N,p)$02 and $\Gamma = G(N,p)$03 copies of $\Gamma = G(N,p)$04, where $\Gamma = G(N,p)$05 and $\Gamma = G(N,p)$06 are the dense extremal minimum and maximum copy counts. This generalises the strictly balanced result of Conlon–Gowers–Samotij–Schacht to all hypergraphs and improves the failure probability to asymptotic optimality.

Sparse pentagon theorem. Combining transference for two structures simultaneously (triangles and $\Gamma = G(N,p)$07s) with the triangle removal lemma, the paper shows that for $\Gamma = G(N,p)$08, with high probability every triangle-free $\Gamma = G(N,p)$09 satisfies $\Gamma = G(N,p)$10, transferring the Erdős pentagon theorem of Grzesik and of Hatami–Hladký–Král'–Norin–Razborov to the sparse setting.

Sparse Ramsey multiplicity. For every $\Gamma = G(N,p)$11-colouring of $\Gamma = G(N,p)$12 with $\Gamma = G(N,p)$13, $\Gamma = G(N,p)$14, there are at least $\Gamma = G(N,p)$15 monochromatic copies of $\Gamma = G(N,p)$16, where $\Gamma = G(N,p)$17 is the dense Ramsey multiplicity constant. This holds regardless of whether $\Gamma = G(N,p)$18 is known.

Sparse Szemerédi multiplicity. For $\Gamma = G(N,p)$19, with probability $\Gamma = G(N,p)$20, every subset of $\Gamma = G(N,p)$21 of size at least $\Gamma = G(N,p)$22 contains at least $\Gamma = G(N,p)$23 $\Gamma = G(N,p)$24-term arithmetic progressions. The machinery applies directly to $\Gamma = G(N,p)$25, avoiding the technical detour through $\Gamma = G(N,p)$26 required by Conlon and Gowers, and achieves the optimal failure probability.

Sparse canonical van der Waerden. The paper gives a short new proof of a recent theorem of Alvarado–Kohayakawa–Morris–Mota–Ortega: for $\Gamma = G(N,p)$27, every colouring of $\Gamma = G(N,p)$28 contains a $\Gamma = G(N,p)$29-term arithmetic progression that is monochromatic or rainbow. The proof demonstrates the handling of multicoloured structures, splitting colour classes and using a probabilistic argument to find many rainbow progressions in the dense model.

Sparse counting lemma. The culminating application is a full sparse counting lemma for hypergraphs: if $\Gamma = G(N,p)$30 is a $\Gamma = G(N,p)$31-complex with $\Gamma = G(N,p)$32-regularity counting (known for $\Gamma = G(N,p)$33 when $\Gamma = G(N,p)$34 is linear and for $\Gamma = G(N,p)$35 in general) and $\Gamma = G(N,p)$36 is an $\Gamma = G(N,p)$37-regular $\Gamma = G(N,p)$38-partition whose top level lies in $\Gamma = G(N,p)$39, then the number of $\Gamma = G(N,p)$40-partite copies of $\Gamma = G(N,p)$41 is $\Gamma = G(N,p)$42, valid for $\Gamma = G(N,p)$43. The lower bound on $\Gamma = G(N,p)$44 is optimal up to constants. The graph case removes the strict-balancing restriction in the counting lemma of Conlon–Gowers–Samotij–Schacht, which had noted that a polylogarithmic loss in $\Gamma = G(N,p)$45 would be required by their methods without providing details.

Limitations and open questions

The paper is candid about several restrictions. Matchings are excluded from the simplified statements, being handled only with a polylogarithmic loss in $\Gamma = G(N,p)$46; the authors describe the resulting regime (a polylogarithmic expected number of edges) as uninterestingly sparse. The $\Gamma = G(N,p)$47 term in the lower bound on $\Gamma = G(N,p)$48 is needed for the Kim–Vu inequality, and at least a single $\Gamma = G(N,p)$49 is needed for the union bounds; the authors expect the high power could be removed by sharper concentration inequalities but did not pursue this, as no known application is affected. The multiple-counts theorem requires $\Gamma = G(N,p)$50 polynomially separated from $\Gamma = G(N,p)$51 and $\Gamma = G(N,p)$52, stronger than the single-structure conditions, though these are typically met in counting applications. The transference principle cannot resolve sparse analogues of dense problems that are themselves open (such as Turán's problem for $\Gamma = G(N,p)$53); it transfers dense answers once they are found. Finally, the paper conjectures that a $\Gamma = G(N,p)$54-complex has $\Gamma = G(N,p)$55-regularity counting exactly when any two $\Gamma = G(N,p)$56-edges intersect in at most $\Gamma = G(N,p)$57 vertices, proving necessity via a standard example and reducing sufficiency to the down-closure case; intermediate values $\Gamma = G(N,p)$58 had not previously been considered.

Conclusion

This paper establishes a generalised transference principle that subsumes the Conlon–Gowers framework and the extremal consequences of Schacht's and the container methods, under the same codegree conditions that govern hypergraph containers. Its distinguishing features are the removal of the strict-balancing restriction from full counting results, asymptotically optimal bounds on both $\Gamma = G(N,p)$59 and the failure probability, and the flexibility to handle structure transfer, localised counting, colours, and multiple structures within a single abstract statement. The proof resolves technical obstructions in the Conlon–Gowers approach through a cleaner norm, randomised splitting, and Kim–Vu-based moment bounds, and the resulting machinery delivers sparse counting lemmas for all graphs and hypergraphs together with a range of Turán-, Ramsey-, and Szemerédi-type corollaries.