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On a Ramsey--Tur\'{a}n variant of Roth's theorem

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Pith's one-line read For a homogeneous linear equation over F_p, solution-free sets with sublinear Cayley independence are small exactly when a nonempty subset of the coefficients sums to zero, completing the Ramsey–Turán analogue of Roth's theorem.

desk verdict A clean, genuinely new Ramsey–Turán variant of Roth's theorem over F_p with a complete classification; the proof is solid and the small blemishes are typo-level. read the letter →

arxiv 2507.22831 v1 pith:3MZYXTLF submitted 2025-07-30 math.CO

Matija Bucić , Micha Christoph , Jaehoon Kim , Hyunwoo Lee , Varun Sivashankar This is my paper · ORCID
classification math.CO MSC 05D1011B3005C2505C35
keywords Ramsey–TurántheoryRoth'stheoremdensityregularityhomogeneouslinearequationsCayleygraphsindependencenumberfinitefieldsadditivecombinatorics
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

This paper asks when a subset $A\subset\mathbb{F}_p$ that contains no solution to a fixed homogeneous linear equation $L$ must be small, after one rules out the usual structured extremal examples by demanding that the Cayley graph generated by $A$ have independence number $o(p)$. The answer is a complete classification: every such solution-free set has size $o(p)$ if and only if some nonempty set of $L$'s coefficients sums to zero. This is a Ramsey–Turán variant of the classical density-regularity theorem for homogeneous linear equations, adding the sublinear-independence condition that the classical extremal examples fail. The degenerate case comes with the quantitative bound $d(L,\varepsilon)\le 100^{k+1}k^3\varepsilon$, and the paper shows this linear dependency is tight for the Schur equation and polynomial in general.

What carries the argument

The engine is a lemma about directed graph systems. A restricted digraph system consists of properly edge-colored digraphs on a common vertex set together with a function assigning each vertex a small set of forbidden colors; a proper rainbow directed path is a path whose $i$-th edge lies in the $i$-th digraph, whose edge colors are all distinct, and whose colors avoid the forbidden sets of the two endpoints. The lemma shows that when each digraph has independence number at most $|V|/(100^{k'}\ell^2)$, such a path of length $k'$ must exist. In the degenerate case the Cayley digraphs generated by coefficient dilations $c_iA$ are the digraphs, colors are the elements of $A$, and the zero-sum subset of coefficients provides a disjoint family of solutions of a smaller subequation; the rainbow path then threads these solutions together into a full solution of $L$.

What would settle it

Take the Schur equation $x+y-z=0$ and fix a small $\varepsilon>0$; Theorem 3.1 says $\limsup_{p\to\infty}D(L,\varepsilon,p)/p\le 100^{4}\cdot27\,\varepsilon$, while Theorem 1.3 says the true order is $\Theta(\varepsilon)$. Any infinite family of primes whose normalized maximum size exceeds that upper bound, or any degenerate equation whose normalized maximum size decays more slowly than linearly in $\varepsilon$, would contradict the quantitative claims; for the non-degenerate direction, checking the paper's explicit construction for $x+y+z=0$ on moderate primes is a finite verification that its independence is $O(p/\log p)$.

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Extended reading notes

Core claim

Let $L:c_1x_1+\cdots+c_kx_k=0$ be a homogeneous linear equation with $k\ge 3$ nonzero integer coefficients, and call $A\subseteq\mathbb{F}_p$ solution-free when no $k$-tuple of distinct elements of $A$ solves $L$. The theorem states that every solution-free $A$ with $\alpha(\mathrm{Cay}_{\mathbb{F}_p}(A))=o(p)$ has $|A|=o(p)$ exactly when $L$ is degenerate, meaning some nonempty subset of its coefficients sums to zero. For non-degenerate equations the paper constructs solution-free sets of linear size whose Cayley graphs still have independence $O(p/\log p)$, so sublinear independence does not force smallness. For degenerate equations the proof yields $d(L,\varepsilon)\le 100^{k+1}k^3\varepsilon$, where $d(L,\varepsilon)$ is the asymptotic maximum density of solution-free sets with independence at most $\varepsilon p$.

Load-bearing premise

The degenerate direction leans on the classical density theorem that every constant-positive-density subset of $\mathbb{F}_p$ contains a solution to any zero-sum equation in at least three variables with distinct entries; if that theorem or its distinct-entry convention failed, the classification would not follow.

Editorial extensions

If this is right

  • For any degenerate equation, the maximum density of a solution-free set with $\alpha(\mathrm{Cay}_{\mathbb{F}_p}(A))\le\varepsilon p$ is at most $100^{k+1}k^3\varepsilon$, so the density tends to zero with $\varepsilon$.
  • For any non-degenerate equation, there is a fixed $\beta>0$ and a solution-free set of size at least $\beta p$ with $\alpha=O(p/\log p)$, which is $o(p)$; hence sublinear independence does not suffice to force smallness unless a zero-sum coefficient subset exists.
  • For the Schur equation $x+y-z=0$, the quantitative rate is $d(L,\varepsilon)=\Theta(\varepsilon)$, so the general linear upper bound cannot be improved in the exponent.
  • For every equation whose coefficients have nonzero total sum but some zero-sum subset, the lower bound is $\varepsilon^{\Theta(1)}$, matching the qualitative classification.
  • A set with $\alpha(\mathrm{Cay}_{\mathbb{F}_p}(A))=o(p)$ is a difference intersector, meeting $B-B$ for every $B$ of positive linear density; the theorem therefore classifies when every difference-intersecting solution-free set is small.

Reading between the lines

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

  • The same classification is plausibly portable to $\mathbb{Z}$ with a difference-intersector condition; the paper notes the translation is routine, so an exact integer analogue would be a direct check rather than a new theorem.
  • The quantitative gap between $O(\varepsilon)$ and $\varepsilon^{\Theta(1)}$ suggests testing intermediate degenerate equations: equations with an isolated zero-sum pair may already force the linear rate, while equations needing larger zero-sum subsets may not.
  • If the independence assumption is strengthened to $o(p/\log p)$ or smaller, the paper's non-degenerate constructions fail, so the classification boundary could shift; whether it does is an open question the paper explicitly raises.
  • The use of sparse high-girth graphs in the lower bound suggests that progress in Ramsey–Turán graph constructions would translate directly into sharper polynomial exponents for non-degenerate equations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 7 minor

Summary. The paper proves a Ramsey-Turán analogue of Roth's theorem for homogeneous linear equations over F_p. The main result (Theorem 1.2) classifies equations for which every solution-free set A with independence number α(Cay_{F_p}(A))=o(p) must have size o(p): this holds exactly when some nonempty subset of the coefficients sums to zero. The degenerate direction is proved with the quantitative bound d(L,ε) ≤ 100^{k+1} k^3 ε, using a new rainbow directed path lemma combined with Roth's theorem; the non-degenerate direction is witnessed by an explicit construction with |A| ≥ βp and α(Cay(A)) = O_L(p/log p). The paper also establishes d(L,ε)=Θ(ε) for Schur's equation and d(L,ε)=ε^{Θ(1)} for every non-degenerate equation possessing a proper zero-sum subset of coefficients.

Significance. If the results hold, the paper gives a clean classification of density-regular homogeneous equations under the Ramsey-Turán independence condition, a natural structural restriction introduced by Erdős and Sárközy. The proof strategy is original: the degenerate case is reduced to a graph-theoretic rainbow path lemma, while the non-degenerate case is handled by explicit constructions that transfer Ramsey graph lower bounds into the additive setting. The paper is careful about the distinct-elements convention in the definition of solution-free sets, and the main classification proof is detailed and self-contained apart from standard tools (Roth's theorem, Caro-Wei, Ramsey graph lower bounds). The quantitative bounds, though probably not optimal except for Schur's equation, give a clear picture of the ε-dependence and raise interesting open problems.

major comments (1)
  1. [Section 4, Theorem 4.4] The proof claims that a graph G with q=t^k vertices and α(G)≤t^{k−1} is guaranteed by Theorem 4.3. With n=t^k, Theorem 4.3 only gives α(G)≤C t^{k−1} log t; the extra log factor invalidates the pigeonhole step α(Cay_Fp(X))≤p/t, because q/t is no longer larger than α(G). The theorem is likely repairable by choosing n≈t^k polylog(t) and absorbing the resulting polylog factors into the exponent C, but as written the proof of d(L,ε)≥ε^C is incomplete.
minor comments (7)
  1. [Section 3.1, Lemma 3.2] In the induction step, the second and fourth bullets refer to 'the color of →v_i v_{i+1} in D_{i+1}' but the edge from v_i to v_{i+1} in the proper rainbow path lies in D_i; the index should be D_i.
  2. [Section 3.2, Theorem 3.3] In the proof of the solution-free claim, the sentence 'assume y_1 is the largest among the y_i's' should read 'assume z_1 is the largest among the z_i's'.
  3. [Section 4, Theorem 4.2] The displayed inclusion 'X_i±X_i ⊆ (8p/9,p) ∪ [0,p/9)' is not literally correct: X_i+X_i is contained in [0,2p/9), while X_i−X_i is contained in (8p/9,p) ∪ [0,p/9). The argument only needs both sets to be disjoint from Y=[p/3,4p/9], so the proof is unaffected, but the inclusion should be stated accurately.
  4. [Section 4, Theorem 4.4] The base-r digit argument is written as if the pairs {a_j,b_j} are distinct, but coefficients c_j can repeat or cancel; the footnote about partial cancellation should be expanded so that the 'each exponent appears in at least two distinct pairs' conclusion is fully justified.
  5. [Section 4, Theorem 4.4] The bound 'at most r q^r' for the size of X_Σ appears too small; a safe bound such as (2|X|)^r would still lead to a polynomial bound for r' after absorbing constants, so the numerical estimate should be corrected or loosened.
  6. [Throughout] Several lemmas are referred to as theorems (e.g., 'Theorem 2.1', 'Theorem 2.2', 'Theorem 3.2' for Lemma 2.1, Lemma 2.2, and Lemma 3.2); consistent numbering would avoid confusion.
  7. [Section 3.1, Theorem 3.1] The repeated application of Roth's theorem to obtain the family S of disjoint solutions is correct because Theorem 1.1 is stated for the distinct-elements notion of solution-free, but a sentence making this contrapositive explicit would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Ramsey–Turán classification is derived from independent external results, not from its own conclusion.

full rationale

The paper's central claim, Theorem 1.2, classifies density-regular homogeneous linear equations under a Ramsey–Turán independence condition. The forward direction (b)->(a) (Theorem 3.1) uses Roth's theorem (Theorem 1.1) as an external black box to find many disjoint solutions of a zero-sum subequation, and then uses a greedy proper-rainbow-path lemma (Lemma 3.2). Roth's theorem is an independent classical result, not the theorem being proved, and the rainbow-path lemma is proved by induction from Caro-Wei degree arguments. The reverse direction (a)->(b) (Theorem 3.3) constructs an explicit linear-size solution-free set with independence number O(p/log p), relying only on elementary modular arithmetic and a clique bound in the Cayley graph. The quantitative lower bounds in Section 4 use external Ramsey graph lower bounds due to Kim and Osthus–Taraz; those results are not self-citations and do not assume Theorem 1.2. The quantity d(L,ε) is defined independently of the theorem's zero-sum-subset condition, so the conclusion is not baked into the definition. The only notable subtlety is the repeated application of Roth's theorem to extract solutions with distinct, pairwise-disjoint elements; this is a standard supersaturation consequence and is not equivalent to the paper's conclusion. Even if the manuscript could be more explicit about that supersaturation step, this is a completeness or correctness concern, not circularity. No load-bearing self-citation, fitted-input-renamed-as-prediction, or definitional equivalence was found.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on Roth's theorem, the Ramsey graph constructions, and elementary graph inequalities. No free parameters are fitted to data.

assumptions (3)
  • standard math Roth's theorem (Theorem 1.1): any subset of F_p of size at least εp contains a solution to any zero-sum homogeneous equation with at least 3 variables, for p large.
    External density classification, used repeatedly in the proof of Theorem 3.1.
  • standard math Caro-Wei bound (Lemma 2.1): a graph with average degree d has independence number at least n/(d+1).
    Used in Lemmas 2.2 and 2.3 to translate independence number conditions into degree conditions.
  • standard math Ramsey graph lower bounds (Theorems 4.1 and 4.3 by Kim and by Osthus-Taraz).
    Used in Section 4 to construct graphs with small independence number and no short cycles, yielding lower bounds on d(L,ε).

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Pith. "Pith review of On a Ramsey--Tur\'{a}n variant of Roth's theorem." pith.science (2026). https://pith.science/paper/3MZYXTLF

@misc{pith2026250722831,
  author       = {Pith},
  title        = {Pith review of: On a Ramsey--Tur\'an variant of Roth's theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3MZYXTLF}},
  note         = {Machine review of arXiv:2507.22831}
}
abstract

A classical theorem of Roth states that the maximum size of a solution-free set of a homogeneous linear equation $\mathcal{L}$ in $\mathbb{F}_p$ is $o(p)$ if and only if the sum of the coefficients of $\mathcal{L}$ is $0$. In this paper, we prove a Ramsey--Tur\'{a}n variant of Roth's theorem, with respect to a natural notion of ``structured'' sets introduced by Erd\H{o}s and S\'ark\"ozy in the 1970's. Namely, we show that the following statements are equivalent: $(a)$ Every solution-free set $A$ of $\mathcal{L}$ in $\mathbb{F}_p$ with $\alpha(\mathrm{Cay}_{\mathbb{F}_p}(A)) = o(p)$ has size $o(p)$. $(b)$ There exists a non-empty \emph{subset} of coefficients of $\mathcal{L}$ with zero sum.

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Reference graph

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