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Sharper Ramsey lower bounds from refined Gaussian estimates

T0 review · 2 major / 2 minor · reviewed 2026-07-04 · grok-4.3

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Pith's one-line read A sharp cumulant generating function bound for truncated Gaussians improves the exponent in lower bounds for off-diagonal Ramsey numbers R(ℓ, Cℓ) by a positive amount for every fixed C > 1.

desk verdict Modest positive exponent improvement for R(ℓ, Cℓ) via sharper CGF bound on truncated Gaussians. read the letter →

arxiv 2605.25843 v2 pith:AVJ56K3B submitted 2026-05-25 math.CO

Qizhong Lin , Lin Niu This is my paper · ORCID
classification math.CO
keywords Ramseynumbersoff-diagonalGaussianrandomgraphscumulantgeneratingfunctionlowerboundstruncatedGaussiansexponentialimprovements
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 establishes a further improvement to the exponential lower bounds for the off-diagonal Ramsey numbers R(ℓ, Cℓ). It does so by replacing the sub-Gaussian tail estimates for truncated Gaussian random variables in the random-graph construction with a more precise bound on their cumulant generating functions. This change produces a strictly better exponent in the lower bound for any fixed C > 1. As C grows large the size of the improvement behaves like Θ(p_C^{-1/2} / log C). The result extends the recent exponential improvements obtained by earlier authors using Gaussian random graphs.

What carries the argument

The sharp cumulant generating function bound for truncated Gaussian random variables, which replaces the earlier sub-Gaussian estimate in the random-graph construction for Ramsey lower bounds.

What would settle it

A direct numerical computation or analytic counterexample showing that the cumulant generating function for a truncated Gaussian exceeds the claimed bound for some parameter values used in the Ramsey construction.

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Extended reading notes

Core claim

By using a sharp cumulant generating function bound for truncated Gaussians in place of sub-Gaussian estimates, the exponent in the lower bound for R(ℓ, Cℓ) can be increased by a strictly positive amount for every fixed C > 1, with the gain asymptotically Θ(p_C^{-1/2}/log C) as C → ∞.

Load-bearing premise

The sharp cumulant generating function bound for truncated Gaussians holds with the stated constants and applies uniformly in the relevant parameter range.

Editorial extensions

If this is right

  • The lower bound exponent improves for every fixed C > 1.
  • The improvement grows like Θ(p_C^{-1/2}/log C) as C tends to infinity.
  • The method applies to the Gaussian random graph model used in recent proofs.
  • Quantitative bounds on R(ℓ, Cℓ) become stronger than those from the prior Gaussian approach.

Reading between the lines

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

  • If the cumulant bound can be sharpened further, additional gains in the Ramsey exponent may be possible.
  • The same refinement might apply to other probabilistic constructions that rely on truncated Gaussians.
  • Testing the cumulant bound numerically for moderate parameter values would confirm the uniform applicability assumed in the proof.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The paper claims a quantitative improvement to lower bounds on the off-diagonal Ramsey numbers R(ℓ, Cℓ) for fixed C > 1 and large ℓ. Building on the Gaussian random-graph constructions of Ma-Shen-Xie and Hunter-Milojević-Sudakov, it replaces a sub-Gaussian tail bound on truncated Gaussians with a sharp cumulant-generating-function estimate, asserting that the exponent in the lower bound increases by a strictly positive amount for every fixed C > 1, with the gain asymptotically Θ(p_C^{-1/2}/log C) as C → ∞.

Significance. If the claimed CGF bound holds uniformly with the stated constants, the work supplies a further, explicitly quantified strengthening of the first exponential improvements over the classical Erdős lower bound for R(ℓ, Cℓ). Such refinements to the analytic estimates in probabilistic constructions are potentially reusable in other extremal problems that rely on truncated-Gaussian or sub-Gaussian tail controls.

major comments (2)
  1. [Proof of the CGF bound and its application in §3–4] The central claim rests on the assertion that the new cumulant-generating-function bound for truncated Gaussians holds uniformly over the truncation thresholds, variances, and edge-probability regimes that appear when the random-graph construction is instantiated for R(ℓ, Cℓ) with large ℓ and arbitrary fixed C > 1. The manuscript must supply an explicit statement of the range of parameters for which the bound is proved and verify that the error terms remain controlled when these parameters are substituted into the Ramsey lower-bound argument; otherwise the asserted Θ(p_C^{-1/2}/log C) gain is not guaranteed.
  2. [Comparison paragraph following the statement of the main theorem] The paper states that the improvement is achieved by replacing the sub-Gaussian estimate with the sharp CGF bound, yet the quantitative comparison between the two estimates (including the precise constants that produce the positive exponent gain) is not displayed in a single location. A direct side-by-side calculation showing how the new bound improves the exponent for a concrete C (e.g., C = 2) would make the gain verifiable.
minor comments (2)
  1. Notation for the truncation level and the parameter p_C should be introduced once and used consistently; currently the same symbol appears with slightly different meanings in the abstract and the construction section.
  2. [Introduction, paragraph 3] The statement 'as C → ∞, the gain is asymptotically Θ(p_C^{-1/2}/log C)' would benefit from an explicit definition of p_C in the introduction rather than only in the technical sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive suggestions. We address each major comment below.

read point-by-point responses

  1. Referee: The manuscript must supply an explicit statement of the range of parameters for which the CGF bound is proved and verify that the error terms remain controlled when these parameters are substituted into the Ramsey lower-bound argument; otherwise the asserted Θ(p_C^{-1/2}/log C) gain is not guaranteed.

    Authors: The proof of the CGF bound in Lemma 3.2 is uniform over the relevant parameter ranges that arise in the construction for any fixed C > 1. Specifically, the bound holds for truncation levels up to O(√log(1/p)), with p in the range used for the random graph model. The error terms are bounded in the derivation of the main lower bound in Section 4. To make this fully explicit as requested, we will add a remark stating the precise parameter ranges and confirming the control of errors in the revision. revision: yes

  2. Referee: The quantitative comparison between the two estimates (including the precise constants that produce the positive exponent gain) is not displayed in a single location. A direct side-by-side calculation showing how the new bound improves the exponent for a concrete C (e.g., C = 2) would make the gain verifiable.

    Authors: We agree that a consolidated comparison would improve readability. In the revised manuscript, we will insert a paragraph after the main theorem that displays the exponent from the sub-Gaussian bound and from the new CGF bound side-by-side for C=2, highlighting the positive gain. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; analytic upgrade is independent of inputs

full rationale

The paper derives a strictly positive exponent improvement for R(ℓ, Cℓ) lower bounds by substituting a sharp CGF bound on truncated Gaussians for the prior sub-Gaussian tail estimate. The abstract and method description present this as a direct analytic refinement with stated constants and uniformity, without any reduction of the claimed Θ(p_C^{-1/2}/log C) gain to a fitted parameter, self-definition, or self-citation chain. No equations or steps are shown that equate the output exponent to the input bound by construction. The argument is self-contained and externally falsifiable via the CGF inequality itself.

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

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The central claim rests on an unverified analytic bound whose details are not supplied.

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Cite this review

Pith. "Pith review of Sharper Ramsey lower bounds from refined Gaussian estimates." pith.science (2026). https://pith.science/paper/AVJ56K3B

@misc{pith2026260525843,
  author       = {Pith},
  title        = {Pith review of: Sharper Ramsey lower bounds from refined Gaussian estimates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AVJ56K3B}},
  note         = {Machine review of arXiv:2605.25843}
}
abstract

Recently, Ma, Shen and Xie broke the Erd\H{o}s barrier for off-diagonal Ramsey numbers $R(\ell,C\ell)$, achieving the first exponential improvement over the classical lower bound for every $C>1$ and sufficiently large $\ell$. Hunter, Milojevi\'{c}, and Sudakov later gave a simplified proof using Gaussian random graphs and obtained better quantitative bounds. In this paper we prove a further improvement, and show that the exponent in the Ramsey lower bound can be increased by a strictly positive amount for every fixed $C>1$; as $C\to\infty$, the gain is asymptotically $\Theta(p_C^{-1/2}/\log C)$. The improvement is achieved by replacing the subgaussian estimate for truncated Gaussians with a sharp cumulant generating function bound.

Figures

Figures reproduced from arXiv: 2605.25843 by the authors.

Figure 1
Figure 1. Comparison of HMS and refined red exponents. The refined red curve [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. New Tower-Type Lower Bounds for Hypergraph Ramsey Numbers

    math.CO 2026-06 unverdicted novelty 7.0 of 10

    Improves r_k(k+1,k+1) > s_3(⌊k/2⌋-2) for k≥6 and proves s_3(k) ≥ (twr_{k-2}(2))^2 for k≥5, yielding r_k(k+1,k+1) > (twr_{⌊k/2⌋-4}(2))^2 for k≥14.

Reference graph

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