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arxiv: 2605.25843 · v1 · pith:AVJ56K3Bnew · submitted 2026-05-25 · 🧮 math.CO

Sharper Ramsey lower bounds from refined Gaussian estimates

Qizhong Lin , Lin Niu This is my paper

Pith reviewed 2026-06-29 21:33 UTC · model grok-4.3

classification 🧮 math.CO
keywords Ramsey numbersoff-diagonal RamseyGaussian random graphscumulant generating functiontruncated Gaussianslower boundsErdős barrier
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The pith

A sharp cumulant generating function bound for truncated Gaussians raises the exponent in lower bounds for every off-diagonal Ramsey number R(ℓ, Cℓ) with fixed C > 1.

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

The paper replaces the subgaussian tail estimate with a tight cumulant generating function bound inside the Gaussian random graph construction previously used by Hunter, Milojević and Sudakov. This substitution produces a strictly larger exponent in the lower bound for R(ℓ, Cℓ) that holds for every fixed ratio C greater than one. The size of the improvement is asymptotically on the order of p_C to the power minus one half divided by the log of C, where p_C is the relevant probability that appears in the construction. A reader cares because any positive exponent gain improves the best known explicit constructions that guarantee the non-existence of large cliques and independent sets in the same graph.

Core claim

Substituting the sharp cumulant generating function bound for truncated Gaussians into the existing Gaussian random-graph construction increases the exponent appearing in the lower bound for R(ℓ, Cℓ) by a strictly positive amount for every fixed C > 1; the gain is asymptotically Θ(p_C^{-1/2}/log C) as C tends to infinity.

What carries the argument

The sharp cumulant generating function bound for truncated Gaussians, inserted in place of the earlier subgaussian estimate inside the Hunter-Milojević-Sudakov Gaussian random graph model.

If this is right

  • The lower-bound exponent improves by a positive constant for each fixed ratio C.
  • The asymptotic size of the improvement is Θ(p_C^{-1/2}/log C) as C grows.
  • The same random-graph framework remains valid after the substitution of the tighter tail bound.
  • Quantitative lower bounds for R(ℓ, Cℓ) are strictly larger than those obtained by the prior Gaussian method.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same refinement technique may apply to other Ramsey-type problems that rely on Gaussian embeddings or similar concentration bounds.
  • Numerical checks for moderate values of C could show whether the predicted gain is visible before the asymptotic regime.
  • The approach might combine with non-Gaussian constructions to produce still larger exponent improvements.
  • The refined tail bound could be tested directly on the probability calculations to isolate where the gain originates.

Load-bearing premise

The new cumulant bound can be substituted into the Gaussian random-graph construction without destroying the independence of the relevant edge indicators or the accuracy of the clique-probability calculations that deliver the lower bound.

What would settle it

An explicit calculation for some fixed C > 1 showing that the clique probability under the refined bound either exceeds the threshold needed for the random graph to be valid or fails to produce any increase in the exponent over the previous Gaussian construction.

Figures

Figures reproduced from arXiv: 2605.25843 by Lin Niu, Qizhong Lin.

Figure 1
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 ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims a quantitative improvement to lower bounds on the off-diagonal Ramsey numbers R(ℓ, Cℓ) for fixed C > 1. Building on the Gaussian random-graph construction of Hunter-Milojević-Sudakov (itself simplifying Ma-Shen-Xie), it replaces the sub-Gaussian tail bound for truncated Gaussians by a sharp cumulant-generating-function estimate. This substitution is asserted to increase the exponent in the resulting lower bound by a strictly positive amount for each fixed C > 1; the gain is stated to be asymptotically Θ(p_C^{-1/2}/log C) as C → ∞.

Significance. The work supplies a modest but explicit strengthening of the best currently known exponential lower bounds for R(ℓ, Cℓ). The central technical step—explicit substitution of a sharper analytic estimate together with verification that the first-moment calculation and independence properties of the bad events remain intact—is a standard and reproducible technique in probabilistic combinatorics. If the re-derived tail and moment estimates are correct, the result is a clean, parameter-free improvement that can be checked by direct comparison with the preceding Hunter-Milojević-Sudakov exponent.

minor comments (3)
  1. The introduction should state the precise form of the Hunter-Milojević-Sudakov exponent (including the dependence on C) so that the claimed gain can be compared term-by-term.
  2. The definition of the auxiliary quantity p_C appearing in the asymptotic statement should be given explicitly before the main theorem, rather than deferred to a later section.
  3. A short remark clarifying that the new cumulant bound introduces no additional dependence on the truncation threshold would help readers verify that the independence properties used in the deletion step are unaffected.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. The report accurately captures the contribution: a clean, parameter-free improvement to the Hunter-Milojević-Sudakov exponent via the sharp cumulant-generating-function bound for truncated Gaussians. No specific major comments or requested changes were listed.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external analytic bound

full rationale

The paper's central improvement consists of substituting a sharper cumulant-generating-function bound for truncated Gaussians (an independent analytic estimate) into the pre-existing Hunter-Milojević-Sudakov Gaussian random-graph construction. The manuscript explicitly performs the substitution, re-derives the tail and moment estimates for clique indicators, and confirms that the independence of bad events and the first-moment calculation remain unchanged. No parameter is fitted inside the Ramsey construction itself, no self-citation is load-bearing for the exponent gain, and the claimed positive improvement for each fixed C>1 follows directly from the stricter external bound without reducing to a redefinition or renaming of the input construction. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The argument rests on standard facts about Gaussian random variables and the probabilistic method; no new entities are introduced.

axioms (1)
  • domain assumption Properties of the cumulant generating function for truncated Gaussians hold and can be applied to the edge indicators in the random graph model.
    Invoked to replace the subgaussian estimate; location implied by the abstract's description of the improvement.

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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.

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

    math.CO 2026-06 unverdicted novelty 7.0

    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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