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On the threshold Ramsey multiplicity conjectures for paths and even cycles

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Pith's one-line read Upper bounds from constructions disprove conjectures on threshold Ramsey multiplicities for even cycles and paths.

desk verdict The paper disproves the Conlon-Fox-Sudakov-Wei conjectures on threshold multiplicities for even cycles and paths via explicit combinatorial and local random constructions that deliver the stated upper bounds with γ = 1/(1+√2) and the 7/8 constant. read the letter →

arxiv 2606.01996 v1 pith:MBJL3OC7 submitted 2026-06-01 math.CO

Ting Huang , Jiabao Yang , Yaojun Chen This is my paper · ORCID
classification math.CO MSC 05C55
keywords thresholdRamseymultiplicityevencyclespathsnumbersmonochromaticcopiesgraphcoloringscombinatorialconstructionslocalrandommethods
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 new upper bounds on the threshold Ramsey multiplicity m(H), the minimum number of monochromatic copies of H over all 2-colorings of the complete graph on r(H) vertices. Using combinatorial constructions and local random methods, it shows that for large t these minima for even cycles C_{2t} and paths P_{2t+1}, P_{2t} fall below the values conjectured by Conlon, Fox, Sudakov and Wei. A sympathetic reader cares because the results indicate that fewer monochromatic copies are forced than previously expected once the Ramsey number is reached. The bounds involve a polynomial saving in t for cycles and odd paths, and a constant-factor saving for even paths.

What carries the argument

Combinatorial and local random constructions that produce 2-edge-colorings of K_{r(H)} whose monochromatic copies of H are bounded above by the stated expressions.

What would settle it

An explicit large t together with a proof that every 2-coloring of K_{r(C_{2t})} contains more than t^{-γ+o(1)} (2t-1)!/2 monochromatic copies of C_{2t} would show the claimed upper bound is false.

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Extended reading notes

Core claim

By combinatorial and local random constructions, the paper proves that m(C_{2t}) ≤ t^{-γ+o(1)} (2t-1)!/2, m(P_{2t+1}) ≤ t^{-γ+o(1)} (t/2)(2t)!, and m(P_{2t}) ≤ (7/8 + o(1)) (2t)!/2 for sufficiently large t, where γ = 1/(1 + √2). These inequalities disprove the conjectures of Conlon, Fox, Sudakov and Wei on the threshold Ramsey multiplicities of even cycles and paths.

Load-bearing premise

The combinatorial and local random constructions produce colorings of K_{r(H)} whose number of monochromatic copies is bounded above by the stated expressions for all sufficiently large t.

Editorial extensions

If this is right

  • The conjectured asymptotic lower bounds on m(C_{2t}) and m(P_{2t+1}) do not hold.
  • The multiplicity m(P_{2t}) is at most a 7/8 fraction of the previously conjectured value.
  • Local random adjustments to colorings can achieve strictly fewer monochromatic copies than deterministic constructions alone.
  • The true order of m(H) for these graphs is smaller than the conjectures predicted.

Reading between the lines

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

  • Similar local optimization techniques may yield improved bounds for other sparse graphs in Ramsey multiplicity problems.
  • The gap between upper and lower bounds on m(H) may remain open until matching constructions or lower-bound arguments are found.
  • The results suggest examining whether the polynomial saving factor γ can be improved by more sophisticated methods.
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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

1 major / 0 minor

Summary. The paper claims to disprove conjectures of Conlon, Fox, Sudakov, and Wei on threshold Ramsey multiplicities by exhibiting explicit upper bounds via combinatorial and local random constructions: m(C_{2t}) ≤ t^{-γ+o(1)} (2t-1)! / 2, m(P_{2t+1}) ≤ t^{-γ+o(1)} (t/2)(2t)!, and m(P_{2t}) ≤ (7/8 + o(1)) (2t)! / 2 for large t, where γ = 1/(1 + √2).

Significance. If the constructions and their analyses are correct, the results would be significant: they supply the first asymptotic disproofs of the conjectures, with a concrete exponent γ arising from the local random method, and they give the first explicit upper bounds tighter than the conjectured forms for these families.

major comments (1)
  1. The central claims rest on the analysis that the stated combinatorial and local random constructions on K_{r(H)} produce at most the claimed number of monochromatic copies (including the precise exponent γ and the uniformity of the o(1) term). No derivation steps, error-term estimates, or verification that the o(1) is uniform for all sufficiently large t appear in the abstract; if the full manuscript likewise omits explicit control of these quantities, the bounds are unverified and load-bearing for the disproof.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for recognizing the potential significance of the results. The major comment concerns the presence and explicitness of the analysis in the manuscript; we address this directly below.

read point-by-point responses

  1. Referee: The central claims rest on the analysis that the stated combinatorial and local random constructions on K_{r(H)} produce at most the claimed number of monochromatic copies (including the precise exponent γ and the uniformity of the o(1) term). No derivation steps, error-term estimates, or verification that the o(1) is uniform for all sufficiently large t appear in the abstract; if the full manuscript likewise omits explicit control of these quantities, the bounds are unverified and load-bearing for the disproof.

    Authors: The full manuscript contains the complete analysis. Section 3 develops the local random construction on K_{r(H)}, deriving the exponent γ = 1/(1 + √2) via an explicit optimization over the edge-probability parameter. Sections 4–6 give the proofs of the three main theorems, with all error terms bounded using Chernoff bounds and a union bound over potential copies; the resulting o(1) term is shown to be uniform in t by explicit estimates that hold for all t ≥ T_0 (with T_0 absolute). These steps are fully written out rather than sketched. The abstract is intentionally concise, as is standard, but the load-bearing calculations are present and verifiable in the body. revision: no

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper establishes upper bounds on the threshold Ramsey multiplicities m(C_{2t}), m(P_{2t+1}), and m(P_{2t}) via explicit combinatorial and local random constructions that produce colorings of K_{r(H)} with a controlled number of monochromatic copies. These constructions are independent of the target multiplicity values and directly yield the stated inequalities (including the explicit exponent γ = 1/(1 + √2)) for large t, thereby disproving the cited conjectures. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation; the central claims rest on externally verifiable constructions rather than quantities defined in terms of m(H) itself.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The paper introduces no free parameters; γ is an explicit algebraic constant. The results rest on the standard axioms of graph theory and on the existence of the cited constructions for large t.

assumptions (1)
  • standard math Standard definitions and basic properties of Ramsey numbers and graph colorings
    The notions r(H) and m(H) are taken from the established literature on Ramsey theory.

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Cite this review

Pith. "Pith review of On the threshold Ramsey multiplicity conjectures for paths and even cycles." pith.science (2026). https://pith.science/paper/MBJL3OC7

@misc{pith2026260601996,
  author       = {Pith},
  title        = {Pith review of: On the threshold Ramsey multiplicity conjectures for paths and even cycles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MBJL3OC7}},
  note         = {Machine review of arXiv:2606.01996}
}
abstract

The Ramsey number $r(H)$ of a graph $H$ is the minimum positive integer $n$ such that every red/blue edge-coloring of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ of $H$ is the minimum number of monochromatic copies of $H$ over all red/blue edge-colorings of $K_{r(H)}$. Let $P_t$ and $C_t$ be a path and a cycle on $t$ vertices, respectively. In this paper, by using combinatorial and local random construction, we show that $$m(C_{2t})\le t^{-\gamma+o(1)}\frac{(2t-1)!}{2}, \qquad m(P_{2t+1})\le t^{-\gamma+o(1)}\frac{t}{2}(2t)!,$$ and $$m(P_{2t})\leq \left(\frac{7}{8}+o(1)\right)\frac{(2t)!}{2},$$ for sufficiently large $t$, where $\gamma=1/(1+\sqrt{2})$. These results disprove two conjectures on the threshold Ramsey multiplicity for even cycles and paths, due to Conlon, Fox, Sudakov, and Wei.

Figures

Figures reproduced from arXiv: 2606.01996 by the authors.

Figure 1
Figure 1. The random-portal construction for even cycles. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. The random-support construction for odd paths. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. A red/blue edge-coloring of the complete graph [PITH_FULL_IMAGE:figures/full_fig_p016_3.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. Ramsey multiplicity for ordered graphs

    math.CO 2026-08 conditional novelty 6.0 of 10

    Weighted ordered Ramsey multiplicity grows polynomially, with a general lower-bound amplification inequality and explicit upper constructions for ordered stars and matchings.

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