Pith. sign in

REVIEW 3 minor 34 references

On the Ramsey classes of random hypergraphs

T0 review · 0 major / 3 minor · reviewed 2026-06-29 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The threshold probability is determined above which a random r-graph H satisfies R(H;s) ⊆ R(Q1,...,Qt) for highly connected fixed Q_i including all completes.

desk verdict The paper pins down the threshold for R(H;s) ⊆ R(Q1,...,Qt) when H is random, via a generalized Graham-Łuczak-Rödl-Ruciński condition on highly connected tuples. read the letter →

arxiv 2605.28472 v1 pith:WZD4DZCH submitted 2026-05-27 math.CO

Dingyuan Liu This is my paper · ORCID
classification math.CO
keywords Ramseyclassesrandomhypergraphsthresholdfunctionshypergraphcoloringequivalencehighlyconnectededgecolorings
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 determines the threshold probability p such that a random r-uniform hypergraph H = H^(r)(n,p) belongs to the Ramsey class R(Q1,...,Qt) whenever it belongs to R(F1,...,Fs), for a broad class of fixed Q tuples that includes complete r-graphs. A sympathetic reader would care because this pins down the point at which random hypergraphs begin forcing the same monochromatic patterns that fixed hypergraphs force under edge colorings. The argument proceeds by generalizing an earlier necessary-and-sufficient condition of Graham, Łuczak, Rödl and Ruciński that applies when the target graphs Q_i are highly connected. As a byproduct the paper also characterizes precisely when two tuples of highly connected r-graphs induce identical Ramsey classes.

What carries the argument

The generalized Graham-Łuczak-Rödl-Ruciński condition, which is necessary and sufficient for the Ramsey-class inclusion R(F1,...,Fs) ⊆ R(Q1,...,Qt) when each Q_i is highly connected.

What would settle it

An explicit random r-graph H with edge probability above the claimed threshold together with an s-edge-coloring that produces no monochromatic copy of any Q_i even though it produces the required monochromatic F_i copies.

Watch

Extended reading notes

Core claim

Our main result determines the threshold for R(H;s) ⊆ R(Q1,...,Qt) where H is the random r-graph and the Q_i are fixed, for a large class of such tuples that includes all complete r-graphs. The proof rests on a generalization of the Graham-Łuczak-Rödl-Ruciński result that supplies a necessary and sufficient condition for R(F1,...,Fs) ⊆ R(Q1,...,Qt) whenever the Q_i are highly connected. As a byproduct we characterize when two tuples of highly connected r-graphs are Ramsey equivalent.

Load-bearing premise

The fixed target hypergraphs Q1 through Qt must be highly connected.

Editorial extensions

If this is right

  • The threshold applies directly when each Q_i is a complete r-graph.
  • Two tuples of highly connected r-graphs induce the same Ramsey class precisely when each tuple satisfies the inclusion condition with respect to the other.
  • Membership of the random hypergraph in the target Ramsey class is completely settled by the generalized connectivity condition once the probability exceeds the threshold.

Reading between the lines

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

  • The same threshold technique may apply to other random hypergraph models that are not uniform.
  • The Ramsey-equivalence characterization supplies a practical test for whether a given highly connected tuple is minimal in its class.
  • Analogous density thresholds could be sought for Ramsey properties in random hypergraphs under vertex colorings rather than edge colorings.
Share X Bluesky LinkedIn Reddit HN

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

0 major / 3 minor

Summary. The paper studies Ramsey classes of random r-graphs: it determines the threshold probability p such that R(H;s) ⊆ R(Q1,...,Qt) holds with high probability for H ~ H^(r)(n,p) and a large class of fixed tuples (Q1,...,Qt), including all complete r-graphs. The proof rests on a generalization of the Graham-Łuczak-Rödl-Ruciński necessary-and-sufficient condition that applies when the Qi are highly connected; a byproduct is a characterization of Ramsey equivalence between two such tuples.

Significance. If the claimed generalization and threshold hold, the work supplies the first explicit thresholds for containment of random-hypergraph Ramsey classes inside fixed ones and a clean equivalence criterion for highly connected tuples. These are concrete, falsifiable statements in an area where most prior results are existential or asymptotic; the generalization itself is a reusable technical tool.

minor comments (3)
  1. The abstract states the main result and the key ingredient but supplies no derivation outline or error-bound discussion; a one-sentence sketch of how the generalized GLRR condition is applied to the random case would improve readability.
  2. Notation for the random hypergraph H^(r)(n,p) and the arrow notation G → (F1,...,Fs) is introduced without an explicit reference to the standard source (e.g., the original Graham et al. paper); adding one citation in §1 would help readers.
  3. The statement of the main threshold result (presumably Theorem 1.1 or 3.1) should explicitly record the range of r,s,t for which the result is proved, matching the opening sentence of the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments appear in the report, so our responses below are accordingly limited. We will incorporate any minor polishing or clarifications in the revised manuscript.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper's main result determines a threshold for R(H;s) ⊆ R(Q1,...,Qt) in random hypergraphs by generalizing an external theorem of Graham, Łuczak, Rödl, and Ruciński on highly connected graphs; this cited result is independent of the present work and is not a self-citation. No equations, fitted parameters, ansatzes, or renamings appear in the abstract or described derivation that reduce the claimed threshold to the paper's own inputs by construction. The byproduct characterization of Ramsey equivalence likewise rests on the external generalization rather than internal self-reference. The derivation chain is therefore self-contained against external benchmarks.

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

Abstract alone gives no explicit free parameters, axioms, or invented entities; the generalization of the Graham-Łuczak-Rödl-Ruciński lemma is invoked without stating its assumptions here.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the Ramsey classes of random hypergraphs." pith.science (2026). https://pith.science/paper/WZD4DZCH

@misc{pith2026260528472,
  author       = {Pith},
  title        = {Pith review of: On the Ramsey classes of random hypergraphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZD4DZCH}},
  note         = {Machine review of arXiv:2605.28472}
}
abstract

Let $r,s,t\geq2$ be integers. For $r$-graphs $G$ and $F_1,\dots,F_s$, we write $G\to(F_1,\dots,F_s)$ if every $s$-edge-coloring of $G$ yields a monochromatic copy of $F_i$ in the $i$-th color for some $1\leq i\leq s$. Let $\mathcal{R}(F_1,\dots,F_s)$ denote the family of all $r$-graphs $G$ with $G\to(F_1,\dots,F_s)$. When $F_1=\dots=F_s=F$, we write $\mathcal{R}(F;s)=\mathcal{R}(F_1,\dots,F_s)$. In this paper, we investigate when $\mathcal{R}(H;s)\subseteq\mathcal{R}(Q_1,\dots,Q_t)$ holds, where $H=H^{(r)}(n,p)$ is a random $r$-graph and $Q_1,\dots,Q_t$ are fixed $r$-graphs. Our main result determines the threshold for a large class of such $Q_1,\dots,Q_t$, including complete $r$-graphs. The key ingredient in our proof is a generalization of a result of Graham, {\L}uczak, R\"odl, and Ruci\'nski, which provides a necessary and sufficient condition for $\mathcal{R}(F_1,\dots,F_s)\subseteq\mathcal{R}(Q_1,\dots,Q_t)$, where $Q_1,\dots,Q_t$ are highly connected. As a byproduct, we characterize when two tuples of highly connected $r$-graphs are Ramsey equivalent.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 3 canonical work pages

  1. [1]

    Axenovich, J

    M. Axenovich, J. Rollin, and T. Ueckerdt, Conditions on Ramsey nonequivalence, Journal of Graph Theory 86 (2017), 159–192

    2017

  2. [2]

    T. F. Bloom and A. Liebenau, Ramsey equivalence of Kn and Kn + Kn−1, Electronic Journal of Combinatorics 25 (2018)

    2018

  3. [3]

    Bollob´ as, Threshold functions for small subgraphs,Mathematical Proceedings of the Cambridge Philosophical Society 90 (1981), 197–206

    B. Bollob´ as, Threshold functions for small subgraphs,Mathematical Proceedings of the Cambridge Philosophical Society 90 (1981), 197–206

    1981

  4. [4]

    Bowtell, R

    C. Bowtell, R. Hancock, and J. Hyde, Proof of the Kohayakawa–Kreuter conjecture for the majority of cases, arXiv:2307.16760 (2023)

    work page arXiv 2023

  5. [5]

    Boyadzhiyska, D

    S. Boyadzhiyska, D. Clemens, P. Gupta, and J. Rollin, Ramsey equivalence for asymmetric pairs of graphs, SIAM Journal on Discrete Mathematics 38 (2024), 55–74

    2024

  6. [6]

    Boyadzhiyska and T

    S. Boyadzhiyska and T. Lesgourgues, On the use of senders for asymmetric tuples of cliques in Ramsey theory, Journal of Combinatorial Theory, Series B 169 (2024), 63–95

    2024

  7. [7]

    Christoph, A

    M. Christoph, A. Martinsson, R. Steiner, and Y. Wigderson, Resolution of the Kohayakawa–Kreuter conjecture, Proceedings of the London Mathematical Society 130 (2025)

    2025

  8. [8]

    Clemens, A

    D. Clemens, A. Liebenau, and D. Reding, On minimal Ramsey graphs and Ramsey equivalence in multiple colours, Combinatorics, Probability and Computing 29 (2020), 537–554

    2020

Show all 34 references
  1. [9]

    Conlon and W

    D. Conlon and W. T. Gowers, Combinatorial theorems in sparse random sets, Annals of Mathematics 184 (2016), 367–454

    2016

  2. [10]

    J. Fox, A. Grinshpun, A. Liebenau, Y. Person, and T. Szab´ o, What is Ramsey-equivalent to a clique?, Journal of Combinatorial Theory, Series B 109 (2014), 120–133

    2014

  3. [11]

    Friedgut, V

    E. Friedgut, V. R¨ odl, and M. Schacht, Ramsey properties of random discrete structures, Random Structures & Algorithms 37 (2010), 407–436

    2010

  4. [12]

    Graham, T

    R. Graham, T. Luczak, V. R¨ odl, and A. Ruci´ nski, Ramsey properties of families of graphs,Journal of Combi- natorial Theory, Series B 86 (2002), 413–419

    2002

  5. [13]

    Gugelmann, R

    L. Gugelmann, R. Nenadov, Y. Person, N. ˇSkori´ c, A. Steger, and H. Thomas, Symmetric and asymmetric Ramsey properties in random hypergraphs, Forum of Mathematics, Sigma 5 (2017). 12

    2017

  6. [14]

    Kohayakawa and B

    Y. Kohayakawa and B. Kreuter, Threshold functions for asymmetric Ramsey properties involving cycles,Random Structures & Algorithms 11 (1997), 245–276

    1997

  7. [15]

    Kohayakawa, M

    Y. Kohayakawa, M. Schacht, and R. Sp¨ ohel, Upper bounds on probability thresholds for asymmetric Ramsey properties, Random Structures & Algorithms 44 (2014), 1–28

    2014

  8. [16]

    Kuperwasser, W

    E. Kuperwasser, W. Samotij, and Y. Wigderson, On the Kohayakawa–Kreuter conjecture, Mathematical Pro- ceedings of the Cambridge Philosophical Society 178 (2025), 293–320

    2025

  9. [17]

    Liebenau, L

    A. Liebenau, L. Mattos, W. Mendon¸ ca, and J. Skokan, Asymmetric Ramsey properties of random graphs in- volving cliques and cycles, Random Structures & Algorithms 62 (2023), 1035–1055

    2023

  10. [18]

    Marciniszyn, J

    M. Marciniszyn, J. Skokan, R. Sp¨ ohel, and A. Steger, Asymmetric Ramsey properties of random graphs involving cliques, Random Structures & Algorithms 34 (2009), 419–453

    2009

  11. [19]

    Sviridenkov, A note on hypergraphs with asymmetric Ramsey properties, arXiv:2605.20949 (2026)

    V. Sviridenkov, A note on hypergraphs with asymmetric Ramsey properties, arXiv:2605.20949 (2026)

    2026 arXiv

  12. [20]

    Morris, Some recent results in Ramsey theory, arXiv:2601.05221 (2026)

    R. Morris, Some recent results in Ramsey theory, arXiv:2601.05221 (2026)

    2026

  13. [21]

    Mousset, R

    F. Mousset, R. Nenadov, and W. Samotij, Towards the Kohayakawa–Kreuter conjecture on asymmetric Ramsey properties, Combinatorics, Probability and Computing 29 (2020), 943–955

    2020

  14. [22]

    Mubayi and A

    D. Mubayi and A. Suk, A survey of hypergraph Ramsey problems, Discrete Mathematics and Applications (2020), 405–428

    2020

  15. [23]

    Nenadov and A

    R. Nenadov and A. Steger, A short proof of the random Ramsey theorem, Combinatorics, Probability and Computing 25 (2016), 130–144

    2016

  16. [24]

    Nenadov, Y

    R. Nenadov, Y. Person, N. ˇSkori´ c, and A. Steger, An algorithmic framework for obtaining lower bounds for random Ramsey problems, Journal of Combinatorial Theory, Series B 124 (2017), 1–38

    2017

  17. [25]

    Neˇ setˇ ril and V

    J. Neˇ setˇ ril and V. R¨ odl, Partitions of finite relational and set systems,Journal of Combinatorial Theory, Series A 22 (1977), 289–312

    1977

  18. [26]

    Neˇ setˇ ril and V

    J. Neˇ setˇ ril and V. R¨ odl, Ramsey theorem for classes of hypergraphs with forbidden complete subhypergraphs, Czechoslovak Mathematical Journal 29 (1979), 202–218

    1979

  19. [27]

    Neˇ setˇ ril and V

    J. Neˇ setˇ ril and V. R¨ odl, The partite construction and Ramsey set systems,Discrete Mathematics 75 (1989), 327–334

    1989

  20. [28]

    Neˇ setˇ ril and V

    J. Neˇ setˇ ril and V. R¨ odl, On Ramsey graphs without bipartite subgraphs,Discrete Mathematics 101 (1992), 223–229

    1992

  21. [29]

    Ramsey, On a problem of formal logic, Proceedings of the London Mathematical Society 2 (1930), 264–286

    F. Ramsey, On a problem of formal logic, Proceedings of the London Mathematical Society 2 (1930), 264–286

    1930

  22. [30]

    R¨ odl and A

    V. R¨ odl and A. Ruci´ nski, Threshold functions for Ramsey properties, Journal of the American Mathematical Society 8 (1995), 917–942

    1995

  23. [31]

    R¨ odl and A

    V. R¨ odl and A. Ruci´ nski, Ramsey properties of random hypergraphs,Journal of Combinatorial Theory, Series A 81 (1998), 1–33

    1998

  24. [32]

    Savery, Chromatic number is Ramsey distinguishing, Journal of Graph Theory 99 (2022), 152–161

    M. Savery, Chromatic number is Ramsey distinguishing, Journal of Graph Theory 99 (2022), 152–161

    2022

  25. [33]

    Szab´ o, P

    T. Szab´ o, P. Zumstein, and S. Z¨ urcher, On the minimum degree of minimal Ramsey graphs,Journal of Graph Theory 64 (2010), 150–164

    2010

  26. [34]

    Thomas, Aspects of games on random graphs, PhD thesis, ETH Zurich, 2013

    H. Thomas, Aspects of games on random graphs, PhD thesis, ETH Zurich, 2013. Karlsruhe Institute of Technology, Englerstraße 2, D-76131 Karlsruhe, Germany Email address : liu@mathe.berlin 13

    2013

Pith tools

Reviewed June 29, 2026 · model on record in the stance chip above.