Marton's conjecture (Polynomial Freiman-Ruzsa) constant

C_18 is the least constant such that, whenever A is a subset of 𝔽_2^n with lvert A+Arvert ≤ Klvert Arvert, then A can be covered by K^C_18+o(1) cosets of a subspace of cardinality at most lvert Arvert, where the limit o(1) is with respect to the limit K → ∞.

Current state (tracked automatically)

Best known bounds for constant 18a: 1.46601 ≤ C ≤ 9. Upper bound ([L2024], A simplified argument giving 11 is also provided); lower bound ([GT2009], Consider A a Hamming ball).

All tracked facts and sources

Fact	Value	Source
Best lower bound with an explicit value (constant 18a)	1.46601 [GT2009] — Consider A a Hamming ball	Tao's optimization constants
Best upper bound with an explicit value (constant 18a)	9 [L2024] — A simplified argument giving 11 is also provided	Tao's optimization constants

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Cite

@misc{cairn-constant-18a-marton-s-conjecture-polynomial-freiman-ruzsa-constant,
  title        = {Marton's conjecture (Polynomial Freiman-Ruzsa) constant},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/constant-18a-marton-s-conjecture-polynomial-freiman-ruzsa-constant}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

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Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

Description of constant

$C_{18}$ is the least constant such that, whenever $A$ is a subset of $F_{2}^{n}$ with $∣ A + A ∣ \leq K ∣ A ∣$ , then $A$ can be covered by $K^{C_{18} + o (1)}$ cosets of a subspace of cardinality at most $∣ A ∣$ , where the limit $o (1)$ is with respect to the limit $K \to \infty$ .

Known upper bounds

Bound	Reference	Comments
$7 + 17 = 11.123 \dots$	[GGMT2025]	Usually reported as $12$
$9$	[L2024]	A simplified argument giving $11$ is also provided

Known lower bounds

Bound	Reference	Comments
$1$	Trivial	Consider $K$ basis vectors
$1.46601$	[GT2009]	Consider $A$ a Hamming ball

Additional comments and links

Conjectured to be finite by Katalin Marton, as recorded in [R1999]. It is the special case of the Polynomial Freiman-Ruzsa (PFR) conjecture when the ambient group is a vector space over the field $F_{2}$ . (The precise formulation of the PFR conjecture in the case of unbounded torsion is still not fully settled.)
Surveys on this problem can be found at [G2005], [G-unpub], and [Lovett2015].

References

[G2005] Green, B. J. Finite field models in additive combinatorics. In: Surveys in Combinatorics 2005, London Math. Soc. Lecture Note Series 327, Cambridge University Press, 2005, 1–27.
[G-unpub] Green, B. J. Notes on the polynomial Freiman–Ruzsa conjecture. Unpublished note available at https://people.maths.ox.ac.uk/greenbj/papers/PFR.pdf
[GGMT2025] Gowers, W. T.; Green, B.; Manners, F.; Tao, T. On a conjecture of Marton. Annals of Mathematics, Second Series, Volume 201 (2025), Issue 2, 515–549. arXiv:2311.05762
[Lovett2015] Lovett, S. An Exposition of Sanders’ Quasi-Polynomial Freiman–Ruzsa Theorem. Theory of Computing Library Graduate Surveys 6 (2015), 1–14.
[L2024] Liao, J.-J. Improved Exponent for Marton's Conjecture in $F_{2}^{n}$ . arXiv:2404.09639 (2024).
[R1999] Ruzsa, I. Z. An analog of Freiman’s theorem in groups. Astérisque 258 (1999), 323–326.
[GT2009] Green, B.; Tao, T. Freiman's theorem in finite fields via extremal set theory. Combin. Probab. Comput. 18 (2009), no. 3, 335-355.

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.