Induced arithmetic removal for partition-regular patterns of complexity 1

V. Gladkova

REVIEW 3 major objections 3 minor 22 references

Induced arithmetic removal for partition-regular patterns of complexity 1

T0 review · 3 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that every partition-regular pattern of complexity 1 over a finite field can be completely eliminated from a colouring by recolouring only a tiny fraction of the space.

desk verdict New and likely true, but a fixable parameter bug in Lemma 5.4 currently blocks the proof of the main theorem; worth a serious referee and a focused revision. read the letter →

arxiv 2412.15170 v3 pith:LPUGAU4W submitted 2024-12-19 math.CO

V. Gladkova This is my paper · ORCID

classification math.CO MSC 11B3005D10

keywords arithmeticremovallemmainducedpartition-regularpatternstruecomplexity1finitefieldsRado'stheoremregularityzerocoset

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that if a pattern of linear equations over a finite field is partition-regular and has true complexity 1, then any colouring of $\mathbb{F}_p^n$ with very few instances of that pattern can be recoloured on a tiny fraction of the points to eliminate every instance, including those involving the zero vector. Earlier induced removal results for such patterns always left a small residue of 'non-generic' instances behind; the author shows that partition-regularity supplies enough flexibility to remove all of them. The proof works by selecting not just one but several regular subcosets inside each large coset, using a quantitative form of Rado's theorem to ensure that the chosen subcosets still solve the pattern's equations. If the theorem is right, it closes the gap between the induced removal lemmas for general complexity-1 patterns and the stronger complete-removal results known for translation-invariant patterns.

What carries the argument

The proof's central device is a two-stage subcoset selection. A strong regularity lemma first partitions $\mathbb{F}_p^n$ into cosets of a coarse subspace $H_1$ and then picks regular cosets of a finer subspace $H_2$ inside each $H_1$-coset, as in [7]. The novel step, Lemma 5.3, selects several cosets $H_3+z_1, \dots, H_3+z_m$ inside the zero coset of $H_2$ at once, with the translations $z_j$ solving the pattern's equation $A z = 0$; this is achieved by applying a quantitative Rado theorem over finite fields to an auxiliary colouring of the coset space. Because a whole consistent family of regular subcosets is chosen, the recolouring can be decided by the colour densities on these representatives without any special handling of the zero coset; partition-regularity guarantees that such a family exists.

What would settle it

Take $p=2$, $r=2$, $\epsilon=2^{-10}$. Then Lemma 5.4 sets $D_0 = \lceil \log_2(\epsilon/8)\rceil = -13$, so $D_1$ need not be large and $p^{-D_1}$ can exceed $\epsilon/4$. Computing this inequality would show the proof does not establish the claimed recolouring bound.

Watch

Extended reading notes

Core claim

The central claim (Theorem 1.5) is: for every $\epsilon>0$, integer $r$, and every partition-regular arithmetic pattern $H$ of complexity 1, there is a $\delta > 0$ such that any $r$-colouring of $\mathbb{F}_p^n$ with $H$-instance density at most $\delta$ can be recoloured on at most an $\epsilon$-fraction of $\mathbb{F}_p^n$ to become completely $H$-free. Unlike the earlier induced removal lemma [7], the recolouring removes all instances, including those containing 0; a non-partition-regular pattern can force a monochromatic solution at 0 in every recolouring, so partition-regularity is exactly the condition that makes complete removal possible.

Load-bearing premise

The proof's guarantee that the recolouring changes only a small fraction of the space depends on the coarse subspace being chosen large enough; the bound stated in Lemma 5.4 does not, as written, ensure this for small epsilon.

Editorial extensions

If this is right

Complete removal holds for all partition-regular patterns of complexity 1, including non-translation-invariant patterns such as rainbow solutions to $x+y+z=w$.
Partition-regularity is exactly the dividing line: for a non-partition-regular pattern, some instances (e.g. at 0) must survive any recolouring, as shown by the paper's Example 1.4.
Corollary 1.7: a subset $A$ of $\mathbb{F}_p^n$ with $A+A$ almost disjoint from the complement is $\epsilon$-close to a subspace.
The theorem extends to removing any finite collection of partition-regular complexity-1 patterns simultaneously.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A corrected parameter choice in the recolouring lemma (taking the coarse-subspace codimension large enough that the unchanged fraction is bounded by $\epsilon/4$) would repair the proof's approximation bound and leave the main theorem intact.
The same multiple-subcoset strategy is the natural candidate for extending complete removal to partition-regular patterns of higher true complexity, although the author notes that the required rank control is not yet available.
If the theorem survives scrutiny, it implies that partition-regularity is the exact dividing line between induced removal lemmas that leave a zero-residue and those that do not, at least in the complexity-1 range.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Main theorem is new and likely true, but the written proof has a parameter bug in Lemma 5.4 that breaks the epsilon-recolouring estimate. I would send it to peer review, with the request that the referee focus on that lemma.

What's genuinely good: Fox-Tidor-Zhao explicitly left non-generic instances, and the translation-invariant result does not cover all partition-regular complexity-1 patterns. Closing that gap is a real contribution. The multiple-subcoset selection scheme is new, and the exposition in Section 3 explains clearly why the zero coset forces the change. The counting lemma in Appendix A is proved independently and looks correct. The paper relies honestly on external results — Green's regularity lemma, Serra-Vena's Rado theorem, FTZ's subcoset selection, Manners's bounds — with no circularity. Citation pattern is fine; the one self-reference to forthcoming work is used only for the discussion of bounds, not as a load-bearing step.

The soft spots, in order:

1. Lemma 5.4 sets D0 = ceil(log_p(epsilon/4r)). For any non-trivial epsilon (epsilon < 4r) this is non-positive, so there is no subspace of that codimension. The proof cannot be instantiated as written. If you ignore that and start with D0 = 0, the final inequality requires p^{-D1} <= epsilon/4, i.e. D1 >= log_p(4/epsilon), whereas the proof only guarantees D1 >= log_p(epsilon/4r). For p=2, r=2, epsilon=1/100 this is the difference between D1 >= 9 and D1 >= -9. The entire epsilon-approximation guarantee rests on this line. The repair looks straightforward — take D0 = ceil(log_p(4/epsilon)) and check the subsequent uses of Proposition 5.1 and Lemma 5.3 still go through — but that repair is absent.

2. Lemma 5.3 applies Rado's theorem with 2r colours, but the auxiliary colouring actually maps into subsets of [r], so it uses 2^r colours. This only affects constants.

3. There is an off-by-m factor in the count of non-regular solutions in the same lemma. Again cosmetic.

These are repair-grade defects, not signs of a confused argument. The core idea is sound and the theorem is probably true. The paper will be useful to people working on arithmetic removal, property testing, and finite-field Ramsey theory, and it would make a good reading-group paper because the gap is instructive.

Recommendation: serious referee, yes. Ask the referee to verify the repaired parameter choice in Lemma 5.4 and the colour-count in Lemma 5.3.

Referee Report

3 major / 3 minor

Summary. The paper claims a complete induced arithmetic removal lemma for partition-regular patterns of complexity 1 over F_p^n: whenever an r-colouring has pattern density at most delta, one can recolour an epsilon-fraction of the space so that no instance of the pattern remains, including instances involving 0. This would extend the Fox--Tidor--Zhao induced removal lemma, which necessarily leaves non-generic instances for non-partition-regular patterns. The proof uses a two-stage subcoset selection scheme, a quantitative Rado theorem over finite fields, a counting lemma for complexity-1 systems, and a recolouring lemma constructed from regular subcosets.

Significance. If the proof is completed, the result is a natural and significant strengthening of the Fox--Tidor--Zhao theorem: it gives fully pattern-free recolourings for all partition-regular complexity-1 patterns, not just recolourings with possible non-generic leftovers. The multiple-subcoset idea is actually interesting and adapts the Alon--Fischer--Krivelevich--Szegedy approach to the arithmetic setting. The paper also contains a self-contained proof of the needed counting lemma and a useful discussion of quantitative bounds, including the polynomial true-complexity bounds of Manners. However, as written, the central proof contains several concrete parameter errors, and the main theorem is not established by the current text. The errors appear repairable, but they are load-bearing.

major comments (3)

[Section 5, Lemma 5.4 (first paragraph and final bound)] The proof defines D0 = ceil(log_p(epsilon/4r)) and takes H0 of codimension D0. For every nontrivial choice of parameters with epsilon < 4r, this D0 is non-positive, so no such subspace exists and the construction cannot be instantiated. Even if one ignores this and starts from D0 = 0, property (i) only gives D1 >= log_p(epsilon/4r), while the final recolouring estimate requires p^{-D1} <= epsilon/4, i.e. D1 >= log_p(4/epsilon). These two lower bounds differ substantially. For example, with p=2, r=2 and epsilon=1/100, the written lower bound is about -9 while the needed lower bound is about 9. Consequently, Lemma 5.4 does not establish that phi' differs from phi in at most epsilon|F_p^n| places, and the epsilon-approximation guarantee of Theorem 1.5 is unproved. The repair is to set D0 = ceil(log_p(4/epsilon)) and then check that the subsequent applications of Proposition 5.1 and Lemma 5.3 still go through; as written, this load-bearing parameter is wrong.
[Section 5, Theorem 5.5 (proof, final lower bound)] Theorem 5.5 applies Lemma 5.4 with epsilon' = min(epsilon, epsilon_count(delta')), where delta' = (epsilon/8r)^m/(2m). Lemma 5.4(iv) then gives alpha_i >= epsilon'/8r, not epsilon/8r. The displayed lower bound, however, uses (epsilon/8r)^m and claims a contradiction with delta < (8 r p^{C_rcl} epsilon^{-1})^{-m}/2. If epsilon' is strictly smaller than epsilon, which is the typical situation because epsilon_count(delta') tends to 0 with delta', the product (epsilon'/8r)^m can be smaller than the error term m delta' = (epsilon/8r)^m/2, in which case the counting lemma gives no positive lower bound at all. Thus the contradiction argument in Theorem 5.5 is not justified as written. The proof needs a coherent parameter choice that keeps the recolouring threshold used in Lemma 5.4 compatible with the counting error delta', for example by choosing delta' in terms of the actual lower bound on alpha_i, or by applying Lemma 5.4 with the original epsilon and choosing delta' so that both epsilon_count(delta') >= epsilon p^{-C_zreg}/8r and m delta' < (epsilon/8r)^m/2 hold.
[Section 5, Lemma 5.3 (proof, regularity of monochromatic solutions)] There are two quantitative gaps in the proof of Lemma 5.3. First, the auxiliary colouring psi takes values in P([r]), so it has up to 2^r colours; Theorem 4.2 should be applied with 2^r colours, not with 2r colours as written. Second, the number of monochromatic solutions that fail property (ii) is at most m epsilon' |F_p^D|^{m-l}, not epsilon' |F_p^D|^{m-l}, since a solution fails if any one of its m coordinates lies in a non-regular coset. With the current choice epsilon' <= crado(p,2r,m)/2, the bound m epsilon' |F_p^D|^{m-l} can exceed the guaranteed number crado |F_p^D|^{m-l} of monochromatic solutions when m > 1. The proof therefore does not establish the existence of a monochromatic solution with all cosets regular. Both issues are local and repairable, for instance by using 2^r colours and taking epsilon' <= crado/(4m), but as written Lemma 5.3 is not fully proved.

minor comments (3)

[Section 5, Proposition 5.1] In the statement of Proposition 5.1, the bound in part (i) is written as D1 <= D2 <= Creg(epsilon, zeta, r, N, d), but N is not defined; this should presumably be r, as in the rest of the statement. There is also a typo in Theorem 2.3, where 'regularity lemm' should be 'regularity lemma'.
[Section 5, Lemma 5.4 (recolouring cost accounting)] In the recolouring step for u != 0, the text says that recolouring low-density colour classes changes at most epsilon/4r + zeta <= epsilon/2r proportion of H1+u. This is the per-colour bound; the total proportion over all r colours is at most sum of the low densities, which is at most r(epsilon/4r + zeta) = epsilon/2. The subsequent line 'zeta + r epsilon/2r <= 3epsilon/4' shows that this is what was intended, but the phrasing should be clarified to avoid an apparent off-by-r error.
[Section 1, Example 1.6] In Example 1.6, the claim that there are at most delta |F_p^n|^3 instances is off by a factor of |F_p^n|: the number of l-tuples is |F_p^n|^4, and fixing one variable in a set of size at most delta|F_p^n|/4 gives at most delta|F_p^n|^4/4 instances. The intended statement is that the density is at most delta/4, not that the number of instances is at most delta |F_p^n|^3.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorem is derived from external regularity, Rado, and counting results, and the counting lemma is proved independently in the appendix.

full rationale

The derivation of Theorem 1.5 is self-contained relative to external, independently established ingredients: Green's arithmetic regularity lemma (Theorem 2.3), the Serra–Vena quantitative Rado theorem (Theorem 4.2), the Fox–Tidor–Zhao subcoset selection (Proposition 5.1), and the counting lemma (Lemma 2.7), the last of which is proved in Appendix A rather than imported. None of these inputs is defined in terms of the target removal statement, and no parameter is fitted to the colourings whose instances are to be removed; the recolouring in Lemmas 5.3 and 5.4 is constructed from the original colouring via regularity and Rado's theorem, not assumed to be pattern-free. The only self-citation, reference [8], concerns a lower-bound discussion in Section 6 and is not load-bearing for the main theorem. The parameter issue in Lemma 5.4 identified by the skeptical reader is a technical correctness concern, not a circularity: it does not reduce the conclusion to an input or rename a fitted quantity as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proof rests entirely on established theorems: Green's regularity lemma, Serra-Vena's quantitative Rado theorem, Fox-Tidor-Zhao's intermediate subcoset selection, the Gowers-Wolf true complexity framework, and Manners's polynomial bounds. The counting lemma is derived in the appendix. No new entities, fitted constants, or ad hoc assumptions are introduced. The only apparent circularity risk would be if Proposition 5.1 encoded the removal conclusion, but it is a purely structural statement about regular partitions. The discovered parameter errors are typos in the framework, not hidden circular dependencies.

assumptions (5)

standard math Green's arithmetic regularity lemma (Theorem 2.3) gives a subspace H of bounded codimension such that P(H) is epsilon-regular for a given list of functions.
Invoked in Lemma 5.3 to produce a partition whose cosets are mostly Fourier-uniform; this is the backbone of all arithmetic removal arguments.
standard math Quantitative Rado theorem over finite fields (Theorem 4.2, Serra-Vena): if the matrix A satisfies Rado's column conditions, then every r-colouring of F_p^D (D >= n_rado) contains at least c_rado(p,r,m) p^{D(m-l)} monochromatic solutions to Az=0.
Used in Lemma 5.3 to find a monochromatic solution to Az=0 in the auxiliary colouring of the coset space; the constants c_rado and n_rado are the quantitative inputs.
standard math Intermediate subcoset selection (Proposition 5.1, restated from Fox-Tidor-Zhao [7, Proposition 3.2]): there are nested subspaces H2 <= H1 of H0 such that non-zero cosets of H2 are epsilon-regular and densities on most subcosets approximate densities on the H1-cosets.
This is the external structural result that provides the H1/H2 hierarchy used in Lemma 5.4; it is not proved in this paper.
standard math True complexity of a linear system and the associated counting lemma (Definition 2.4 and Lemma 2.7): for complexity-1 systems, if cosets are consistent and epsilon-regular, Lambda_L is close to the product of the averages.
The counting lemma is proved in Appendix A from the telescoping identity and the definition of true complexity; it is the quantitative engine for boosting one surviving instance into many.
standard math Rado's theorem in abelian groups (Deuber, Bergelson-Deuber-Hindman) characterises partition regularity of the matrix K(L) by Rado's column conditions, used to define partition-regular linear systems in Section 4.
Bridges the arithmetic pattern definition to the matrix form to which the quantitative Rado theorem applies.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Induced arithmetic removal for partition-regular patterns of complexity 1." pith.science (2026). https://pith.science/paper/LPUGAU4W

@misc{pith2026241215170,
  author       = {Pith},
  title        = {Pith review of: Induced arithmetic removal for partition-regular patterns of complexity 1},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LPUGAU4W}},
  note         = {Machine review of arXiv:2412.15170}
}

read the original abstract

In 2019, Fox, Tidor and Zhao (arXiv:1911.03427) proved an induced arithmetic removal lemma for linear patterns of complexity 1 in vector spaces over a fixed finite field. With no further assumptions on the pattern, this induced removal lemma cannot guarantee a fully pattern-free recolouring of the space, as some `non-generic' instances must necessarily remain. On the other hand, Bhattacharyya et al. (arXiv:1212.3849) showed that in the case of translation-invariant patterns, it is possible to obtain recolourings that eliminate the given pattern completely, with no exceptions left behind. This paper demonstrates that such complete removal can be achieved for all partition-regular arithmetic patterns of complexity 1.

Figures

Figures reproduced from arXiv: 2412.15170 by the authors.

**Figure 1.** The sets A (left) and A′ (right) depicted as hashed areas. Blue squares correspond to regular cosets after an application of the arithmetic regularity lemma. A crucial property of A′ is that if a coset contains at least one element of A′ , then A must both be Fourier-uniform and have high density on this coset. As a result, if there were a single instance of L in A′ , we would be able to deduce that there are many i… view at source ↗

**Figure 2.** An instance of L in A′ would imply many instances of L in A as a consequence of Lemma 2.7. A set A may equivalently be viewed as a 2-colouring of F n p given by 1A, and the modified set A′ as an L-free recolouring. In this way, Theorem 3.1 is a special case of the induced [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** The strategy of Fox, Tidor, and Zhao [7] for the induced arithmetic removal of complexity-1 patterns, depicted for a 2-colouring. Instances of the pattern which contain 0 may remain. In order to make sure that the chosen subcosets preserve consistency, Fox, Tidor and Zhao [7] require that these subcosets themselves form a subspace. Of course, this leaves no choice for the subcoset representing the zero coset H1, as … view at source ↗

Figures from the paper (1 more)

**Figure 5.** Figure 5: An auxiliary colouring ψ arising in the proof of Lemma 5.3 from a 3-colouring ϕ : F n p → {r, b, g}. Due to a large proportion of regular cosets (shown in blue), we can find a monochromatic solution consisting entirely of regular cosets. On the other hand, the number o…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 21 canonical work pages

[1]

Efficient testing of large graphs

N. Alon, E. Fischer, M. Krivelevich, and M. Szegedy. “Efficient testing of large graphs”. In: Combinatorica 20.4 (2000), pp. 451–476. REFERENCES 19

work page 2000
[2]

Rado’s Theorem for finite fields

V. Bergelson, W. A. Deuber, and N. Hindman. “Rado’s Theorem for finite fields”. In: Sets, graphs and numbers. Vol. 60. Colloq. Math. Soc. J´ anos Bolyai. J´ anos Bolyai Math. Soc., Budapest, 1992, pp. 77–88

work page 1992
[3]

Every locally characterized affine-invariant property is testable

A. Bhattacharyya, E. Fischer, H. Hatami, P. Hatami, and S. Lovett. “Every locally characterized affine-invariant property is testable”. In: Proceedings of the 2013 ACM Symposium on Theory of Computing . ACM, New York, 2013, pp. 429–435

work page 2013
[4]

A unified framework for testing linear-invariant properties

A. Bhattacharyya, E. Grigorescu, and A. Shapira. “A unified framework for testing linear-invariant properties”. In: Random Structures Algorithms 46.2 (2015), pp. 232– 260

work page 2015
[5]

Graph removal lemmas

D. Conlon and J. Fox. “Graph removal lemmas”. In: Surveys in combinatorics 2013 . Vol. 409. London Math. Soc. Lecture Note. Cambridge Univ. Press, 2013, pp. 1–49

work page 2013
[6]

Partition theorems for abelian groups

W. Deuber. “Partition theorems for abelian groups”. In: J. Combinatorial Theory Ser. A 19 (1975), pp. 95–108

work page 1975
[7]

Induced arithmetic removal: complexity 1 patterns over finite fields

J. Fox, J. Tidor, and Y. Zhao. “Induced arithmetic removal: complexity 1 patterns over finite fields”. In: Israel J. Math. 248.1 (2022), pp. 1–38

work page 2022
[8]

A lower bound for the strong arithmetic regularity lemma

V. Gladkova. “A lower bound for the strong arithmetic regularity lemma”. forthcoming

work page

Show all 22 references

[9]

A new proof of Szemer´ edi’s theorem

W. T. Gowers. “A new proof of Szemer´ edi’s theorem”. In: Geom. Funct. Anal. 11.3 (2001), pp. 465–588

2001
[10]

Decompositions, approximate structure, transference, and the Hahn- Banach theorem

W. T. Gowers. “Decompositions, approximate structure, transference, and the Hahn- Banach theorem”. In: Bull. Lond. Math. Soc. 42.4 (2010), pp. 573–606

2010
[11]

The true complexity of a system of linear equations

W. T. Gowers and J. Wolf. “The true complexity of a system of linear equations”. In: Proc. Lond. Math. Soc. (3) 100.1 (2010), pp. 155–176

2010
[12]

Linear forms and higher-degree uniformity for functions on Fn p

W. T. Gowers and J. Wolf. “Linear forms and higher-degree uniformity for functions on Fn p ”. In: Geom. Funct. Anal. 21.1 (2011), pp. 36–69

2011
[13]

A Szemer´ edi-type regularity lemma in abelian groups, with applications

B. Green. “A Szemer´ edi-type regularity lemma in abelian groups, with applications”. In: Geom. Funct. Anal. 15.2 (2005), pp. 340–376

2005
[14]

Fourier uniformity on subspaces

B. Green and T. Sanders. “Fourier uniformity on subspaces”. pre-print arXiv:1510.08739. 2015

2015 arXiv
[15]

Montr´ eal notes on quadratic Fourier analysis

B. Green. “Montr´ eal notes on quadratic Fourier analysis”. In: Additive combinatorics. Vol. 43. CRM Proc. Lecture Notes. Amer. Math. Soc., Providence, RI, 2007, pp. 69– 102

2007
[16]

An improved lower bound for arithmetic regularity

K. Hosseini, S. Lovett, G. Moshkovitz, and A. Shapira. “An improved lower bound for arithmetic regularity”. In: Math. Proc. Cambridge Philos. Soc. 161.2 (2016), pp. 193– 197

2016
[17]

True complexity and iterated Cauchy–Schwarz

F. Manners. “True complexity and iterated Cauchy–Schwarz”. pre-print arXiv:2109.05731. 2021

2021 arXiv
[18]

Studien zur Kombinatorik

R. Rado. “Studien zur Kombinatorik”. In: Math. Z. 36.1 (1933), pp. 424–470. 20 REFERENCES

1933
[19]

On the number of monochromatic solutions of integer linear systems on abelian groups

O. Serra and L. Vena. “On the number of monochromatic solutions of integer linear systems on abelian groups”. In: European J. Combin. 35 (2014), pp. 459–473

2014
[20]

Regular partitions of graphs

E. Szemer´ edi. “Regular partitions of graphs”. In: Probl` emes combinatoires et th´ eorie des graphes. Vol. 260. Colloq. Internat. CNRS. CNRS, 1978, pp. 399–401

1978
[21]

Testing linear-invariant properties

J. Tidor and Y. Zhao. “Testing linear-invariant properties”. In: SIAM J. Comput. 51.4 (2022), pp. 1230–1279. Appendix A. Proof of the counting lemma This appendix details the proof of the counting lemma stated in the introduction. Recall that ϵcount was defined alongside true ...

2022
[22]

For any functions f1,

Let H ⩽ Fn p be a subspace of codimension d. For any functions f1, . . . , fm : Fn p → [−1, 1] and any c1, . . . , cm ∈ Fn p , if cosets H + c1, . . . , H+ cm are consistent with L and ϵ-regular for f1, . . . , fm, then ΛL(f11 H+c1, . . . , fm1 H+cm) − p−d(m−rank(L)) mY i=1 αi...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Efficient testing of large graphs

N. Alon, E. Fischer, M. Krivelevich, and M. Szegedy. “Efficient testing of large graphs”. In: Combinatorica 20.4 (2000), pp. 451–476. REFERENCES 19

work page 2000

[2] [2]

Rado’s Theorem for finite fields

V. Bergelson, W. A. Deuber, and N. Hindman. “Rado’s Theorem for finite fields”. In: Sets, graphs and numbers. Vol. 60. Colloq. Math. Soc. J´ anos Bolyai. J´ anos Bolyai Math. Soc., Budapest, 1992, pp. 77–88

work page 1992

[3] [3]

Every locally characterized affine-invariant property is testable

A. Bhattacharyya, E. Fischer, H. Hatami, P. Hatami, and S. Lovett. “Every locally characterized affine-invariant property is testable”. In: Proceedings of the 2013 ACM Symposium on Theory of Computing . ACM, New York, 2013, pp. 429–435

work page 2013

[4] [4]

A unified framework for testing linear-invariant properties

A. Bhattacharyya, E. Grigorescu, and A. Shapira. “A unified framework for testing linear-invariant properties”. In: Random Structures Algorithms 46.2 (2015), pp. 232– 260

work page 2015

[5] [5]

Graph removal lemmas

D. Conlon and J. Fox. “Graph removal lemmas”. In: Surveys in combinatorics 2013 . Vol. 409. London Math. Soc. Lecture Note. Cambridge Univ. Press, 2013, pp. 1–49

work page 2013

[6] [6]

Partition theorems for abelian groups

W. Deuber. “Partition theorems for abelian groups”. In: J. Combinatorial Theory Ser. A 19 (1975), pp. 95–108

work page 1975

[7] [7]

Induced arithmetic removal: complexity 1 patterns over finite fields

J. Fox, J. Tidor, and Y. Zhao. “Induced arithmetic removal: complexity 1 patterns over finite fields”. In: Israel J. Math. 248.1 (2022), pp. 1–38

work page 2022

[8] [8]

A lower bound for the strong arithmetic regularity lemma

V. Gladkova. “A lower bound for the strong arithmetic regularity lemma”. forthcoming

work page