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arxiv: 2605.28793 · v3 · pith:BN56EF6Lnew · submitted 2026-05-27 · 🧮 math.CO

Off-diagonal Ramsey numbers

Domagoj Brada\v{c} This is my paper

Pith reviewed 2026-06-29 11:17 UTC · model grok-4.3

classification 🧮 math.CO
keywords Ramsey numbersoff-diagonal Ramsey numberslower boundscliquesindependent setsgraph theory
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The pith

For fixed s at least 3, r(s,k) is at least on the order of k to the s-1 over (log k) to the 2s-4.

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

The paper proves a lower bound for the off-diagonal Ramsey number r(s,k) when s is fixed at least 3 and k grows large. It establishes that r(s,k) grows at least like k to the power s-1 divided by (log k) to the power 2s-4. This nearly matches the upper bound proved by Erdős and Szekeres in 1935. For s at least 5 the new bound improves the long-standing Spencer lower bound of roughly k to the (s+1)/2 power.

Core claim

We prove that for any fixed s ≥ 3 and k tending to infinity, the off-diagonal Ramsey numbers satisfy r(s, k) ≥ Ω (k^{s-1} / (log k)^{2s-4}), which matches, up to polylogarithmic factors, the upper bound established over 90 years ago by Erdős and Szekeres. For s ≥ 5, this improves the best known lower bound of the form r(s, k) ≥ k^{(s+1)/2 + o(1)}.

What carries the argument

A combinatorial construction producing a graph on the claimed number of vertices with neither an s-clique nor a k-independent set.

If this is right

  • The gap between lower and upper bounds for r(s,k) shrinks to only polylogarithmic factors in k.
  • For every fixed s at least 5 the previous polynomial exponent (s+1)/2 is replaced by the larger exponent s-1.
  • The result holds uniformly for all fixed s at least 3 as k tends to infinity.

Where Pith is reading between the lines

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

  • The same style of construction might extend to related Ramsey quantities such as multicolor or hypergraph versions.
  • An explicit version of the construction could yield new algorithmic results on finding large cliques or independent sets.

Load-bearing premise

A graph exists on that many vertices with no s-clique and no k-independent set.

What would settle it

An explicit graph on fewer vertices with neither an s-clique nor a k-independent set, or a proof that no such graph exists on the claimed number of vertices.

Figures

Figures reproduced from arXiv: 2605.28793 by Domagoj Brada\v{c}.

Figure 1
Figure 1. Figure 1: The forbidden structure H5. The vertical green edges represent edges in F and the diagonal red edges represent edges in G. Lemma 2.6 (e.g. [5, Corollary 9.2.5]). Let G be an (n, d, λ)-graph and let A, B ⊆ V (G). Then, [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
read the original abstract

For positive integers $s$ and $k$, the Ramsey number $r(s,k)$ is the minimum integer $n$ such that any graph on $n$ vertices contains a clique of size $s$ or an independent set of size $k$. We prove that for any fixed $s \ge 3$ and $k$ tending to infinity, the off-diagonal Ramsey numbers satisfy \[ r(s, k) \ge \Omega \left(\frac{k^{s-1}}{(\log k)^{2s-4}} \right), \] which matches, up to polylogarithmic factors, the upper bound established over 90 years ago by Erd\H{o}s and Szekeres. For $s \ge 5,$ this improves the best known lower bound of the form $r(s, k) \ge k^{\frac{s+1}{2} + o(1)}$ which was first established by Spencer in 1977 and has since only seen polylogarithmic improvements.

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

1 major / 2 minor

Summary. The paper proves that for fixed s ≥ 3 and k → ∞ the off-diagonal Ramsey number satisfies r(s,k) ≥ Ω(k^{s-1}/(log k)^{2s-4}), matching the Erdős–Szekeres upper bound up to polylog factors and improving the Spencer 1977 lower bound of k^{(s+1)/2 + o(1)} for s ≥ 5. The proof proceeds via an iterative semi-random construction that produces a graph on the claimed number of vertices with no K_s and no independent set of size k.

Significance. If the central construction and its analysis hold, the result is a major advance in Ramsey theory: it nearly closes the gap for off-diagonal numbers after decades of only polylogarithmic progress. The manuscript supplies an explicit combinatorial argument rather than a reduction to prior bounds.

major comments (1)
  1. [§3] §3 (Construction and analysis): The bound on Pr[a fixed k-set remains independent] is obtained via a product over rounds that treats edge decisions as essentially independent. Vertex deletions in prior rounds correlate the remaining edge probabilities; the manuscript does not quantify the resulting multiplicative error. If this error is (log k)^{Ω(1)}, the union bound over inom{n}{k} sets fails at the claimed n, reducing the exponent back toward the classical regime.
minor comments (2)
  1. [Introduction] The abstract states the result but the introduction could more explicitly contrast the new exponent s-1 with the prior (s+1)/2 threshold.
  2. [§2] Notation for the semi-random process (e.g., the parameter controlling deletion probability per round) is introduced without a consolidated table of constants.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and for recognizing the significance of our result on off-diagonal Ramsey numbers. We address the major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (Construction and analysis): The bound on Pr[a fixed k-set remains independent] is obtained via a product over rounds that treats edge decisions as essentially independent. Vertex deletions in prior rounds correlate the remaining edge probabilities; the manuscript does not quantify the resulting multiplicative error. If this error is (log k)^{Ω(1)}, the union bound over \binom{n}{k} sets fails at the claimed n, reducing the exponent back toward the classical regime.

    Authors: The referee correctly identifies a point that requires more explicit justification. In the semi-random process, we control the deletion probabilities so that the expected number of deletions affecting a fixed set is small. The correlation induced is bounded by a factor of exp(O((log log k)/log k)) per round or better, resulting in a total multiplicative factor of (log k)^{o(1)} over all rounds, which can be absorbed into the existing (log k)^{2s-4} term without changing the asymptotic. However, to address the concern directly, we will add a detailed calculation of this error term in a revised §3, making the analysis self-contained. revision: partial

Circularity Check

0 steps flagged

New probabilistic lower-bound construction is self-contained and independent of self-citations or fitted inputs

full rationale

The paper presents a direct combinatorial/probabilistic construction establishing the stated lower bound on r(s,k). No step reduces by definition or fitting to the target quantity; the cited historical results (Erdős-Szekeres, Spencer) are external upper/prior lower bounds and do not form a load-bearing self-citation chain. The derivation chain therefore contains no self-definitional, fitted-prediction, or uniqueness-imported circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper is a proof of an inequality in Ramsey theory; no free parameters or invented entities are apparent from the abstract. The result builds on standard mathematical foundations.

axioms (2)
  • standard math The definition of Ramsey number r(s,k) as the minimal n such that any graph on n vertices contains a clique of size s or an independent set of size k.
    Invoked in the abstract's opening sentence.
  • domain assumption Asymptotics as k tends to infinity with s fixed.
    The bound is stated explicitly in this regime.

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Forward citations

Cited by 2 Pith papers

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    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.

  2. (Auto)formalization is supposed to be easy: Trellis process semantics for spelling out rigorous proofs

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    Trellis applies process semantics to guide generalist LLM agents through incremental refinement of proofs for reliable Lean autoformalization, shown via a Ramsey theory example.

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    ≤ s 2 n−s s−2 −1 . Rearranging, we have (1 +o(1)) s 2 ns−2/(s−2)!≤p −( s 2), implying n≤(1 +o(1)) s e ·(1/p) (s+1)/2+1/(s−2). Completely analogously, we obtain n≤(1 +o(1)) s e ·(1/(1−p)) (k+1)/2+1/(k−2), where we used thatk= (1 +o(1))s. Together, these clearly implyp∈[1/4,3/4], so we may drop the terms2/(s−2)and2/(k−2)from the previous equations to get n≤...