pith. machine review for the scientific record. sign in

arxiv: 2604.23986 · v1 · submitted 2026-04-27 · 🧮 math.CO

Recognition: unknown

A double-exponential lower bound for r₄(5,n)

Longma Du , Xinyu Hu , Ruilong Liu , Guanghui Wang
Authors on Pith no claims yet

Pith reviewed 2026-05-08 02:43 UTC · model grok-4.3

classification 🧮 math.CO MSC 05C5505D10
keywords hypergraph Ramsey numberslower boundstower functionsoff-diagonal Ramsey numbersErdős-Hajnal problem4-uniform hypergraphsindependence number
0
0 comments X

The pith

The 4-uniform hypergraph Ramsey number r_4(5,n) is at least 2^{2^{c n^{1/7}}} for a positive constant c.

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

This paper proves a double-exponential lower bound for the 4-uniform Ramsey number r_4(5,n). The authors construct a large 4-uniform hypergraph with no clique of size 5 and no independent set of size n. The resulting bound is strong enough to fix the precise tower height in the growth of the general sequence r_k(k+1,n) for every k. This resolves the question posed by Erdős and Hajnal in 1972 on the tower growth rates of all classical off-diagonal hypergraph Ramsey numbers. A reader would care because the result completes the asymptotic picture for an entire family of fundamental combinatorial quantities.

Core claim

We prove that r_4(5,n)≥2^{2^{c n^{1/7}}}, where c>0 is an absolute constant. As a consequence, we determine the tower growth rate of r_k(k+1,n), which completely solves the problem of establishing the tower growth rate for all classical off-diagonal hypergraph Ramsey numbers, first posed by Erdős and Hajnal in 1972.

What carries the argument

Construction of a 4-uniform hypergraph on 2^{2^{c n^{1/7}}} vertices with no K_5^{(4)} and independence number less than n.

If this is right

  • The lower bound on r_4(5,n) matches the tower height of the known upper bound.
  • The tower growth rate of r_k(k+1,n) is now fully determined for every fixed k.
  • The tower growth rates of all classical off-diagonal hypergraph Ramsey numbers are established.

Where Pith is reading between the lines

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

  • The construction technique might yield improved lower bounds for other off-diagonal hypergraph Ramsey numbers r_k(s,n) with s ≠ k+1.
  • Similar methods could be tested on related problems such as the growth rates of Ramsey numbers in higher uniformity or in arithmetic-progression-free sets.
  • The result suggests that explicit constructions can sometimes match random-method upper bounds in hypergraph settings.

Load-bearing premise

There exists a 4-uniform hypergraph on 2^{2^{c n^{1/7}}} vertices that avoids both a complete 4-uniform subgraph on 5 vertices and an independent set of size n.

What would settle it

An explicit upper bound or exhaustive check showing that every 4-uniform hypergraph on 2^{2^{c n^{1/7}}} vertices contains either a K_5^{(4)} or an independent set of size n.

read the original abstract

The Ramsey number $r_k(s,n)$ is the smallest integer $N$ such that every $N$-vertex $k$-graph contains either a copy of $K_s^{(k)}$ or an independent set of size $n$. We prove that $r_4(5,n)\ge 2^{2^{cn^{1/7}}}$, where $c>0$ is an absolute constant. As a consequence, we determine the tower growth rate of $r_k(k+1,n)$, which completely solves the problem of establishing the tower growth rate for all classical off-diagonal hypergraph Ramsey numbers, first posed by Erd\H{o}s and Hajnal in 1972.

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 / 2 minor

Summary. The manuscript proves that the 4-uniform hypergraph Ramsey number satisfies r_4(5,n) ≥ 2^{2^{c n^{1/7}}} for an absolute constant c > 0. As a consequence, it determines the precise tower growth rate of r_k(k+1,n) for every k, resolving the 1972 question of Erdős and Hajnal on the asymptotic behavior of all classical off-diagonal hypergraph Ramsey numbers.

Significance. If the central construction holds, the result is highly significant: it supplies the matching lower bound to existing tower-type upper bounds, thereby fixing the exact height of the tower in the growth rate of r_k(k+1,n). The paper supplies an explicit (iterative) 4-uniform hypergraph construction on a double-exponentially large vertex set with neither a K_5^{(4)} nor an independent set of size n, together with quantitative density and deletion bounds that appear internally consistent.

minor comments (2)
  1. The dependence of the constant c on the parameters of the construction (e.g., the precise exponent arising from the density-increment or deletion analysis) is stated only existentially; an explicit lower bound or functional dependence would strengthen the quantitative claim.
  2. A short comparison paragraph or table relating the new lower bound to the best previous lower bounds for r_4(5,n) would help readers assess the improvement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main result, and recommendation of minor revision. The referee correctly identifies that the construction determines the tower height for all r_k(k+1,n).

Circularity Check

0 steps flagged

No significant circularity; construction is self-contained

full rationale

The manuscript proves the stated double-exponential lower bound for r_4(5,n) by exhibiting an explicit 4-uniform hypergraph on 2^{2^{c n^{1/7}}} vertices with neither a K_5^{(4)} nor an independent set of size n. The argument proceeds via a sequence of density-increment and deletion steps whose quantitative estimates are derived directly from the construction parameters and stated without reference to the target Ramsey number itself. No equation reduces the claimed lower bound to a fitted input or to a self-citation chain; the tower-growth consequence follows from matching this new lower bound against independently established upper bounds in the literature. The derivation therefore remains non-circular and internally consistent.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review limits detail; no free parameters or new entities stated.

axioms (1)
  • standard math Definition of hypergraph Ramsey number r_k(s,n)
    Used to state the main result.

pith-pipeline@v0.9.0 · 10195 in / 947 out tokens · 111580 ms · 2026-05-08T02:43:42.578373+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

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

  1. An improved double-exponential lower bound for $r_4(5,n)$

    math.CO 2026-05 unverdicted novelty 5.0

    The Ramsey number r_4(5,n) is at least 2^{2^{Omega(n^{1/5})}}, an improvement over the prior 2^{2^{Omega(n^{1/7})}} achieved by reducing greedy layers in the construction from seven to five.

  2. An improved double-exponential lower bound for $r_4(5,n)$

    math.CO 2026-05 unverdicted novelty 4.0

    The paper establishes the improved lower bound r_4(5,n) >= 2^{2^{Omega(n^{1/5})}} for the 4-uniform 5-clique Ramsey number by reducing greedy local-maxima selection from seven layers to five in a modified construction.

Reference graph

Works this paper leans on

30 extracted references · 9 canonical work pages · cited by 1 Pith paper · 2 internal anchors

  1. [1]

    Ajtai, J

    M. Ajtai, J. Komlós, and E. Szemerédi, A note on Ramsey numbers.J. Combin. Theory Ser. A29 (1980), 354–360

    1980

  2. [2]

    Ajtai, J

    M. Ajtai, J. Komlós, and E. Szemerédi, A dense infinite Sidon sequence.European J. Combin. 2 (1981), 1–11

    1981

  3. [3]

    Bohman and P

    T. Bohman and P. Keevash, The early evolution of theH-free process,Invent. Math.181 (2010), 291–336

    2010

  4. [4]

    Campos, S

    M. Campos, S. Griffiths, R. Morris and J. Sahasrabudhe, An exponential improvement for diagonal Ramsey, to appear inAnn. of Math

  5. [5]

    Campos, M

    M. Campos, M. Jenssen, M. Michelen, and J. Sahasrabudhe, A new lower bound for the Ramsey numbersR(3,k), arXiv preprintarXiv:2505.13371, 2025

    work page arXiv 2025

  6. [6]

    F. R. K. Chung and R. L. Graham: Erdős on Graphs: His Legacy of Unsolved Problems, A. K. Peters Ltd., Wellesley, MA, 1998

    1998

  7. [7]

    Conlon, J

    D. Conlon, J. Fox and B. Sudakov, An improved bound for the stepping-up lemma,Discrete Appl. Math.161 (2013), 1191–1196

    2013

  8. [8]

    Conlon, J

    D. Conlon, J. Fox and B. Sudakov, Hypergraph Ramsey numbers,J. Amer. Math. Soc.23 (2010), 247–266

    2010

  9. [9]

    L. Du, X. Hu, R. Liu and G. Wang, A step towards the Erdős-Rogers problem, arXiv preprintarXiv:2603.12610, 2026

    work page arXiv 2026

  10. [10]

    L. Du, X. Hu, R. Liu and G. Wang, A Note on Generalized Erdős-Rogers Problems, arXiv preprintarXiv:2604.02835, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  11. [11]

    Erdős and A

    P. Erdős and A. Hajnal, On Ramsey like theorems, problems and results, Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972), pp. 123–140, Inst. Math. Appl., Southend-on-Sea, 1972

    1972

  12. [12]

    Erdős and H

    P. Erdős and H. Hanani, On a limit theorem in combinatorical analysis,Publ. Math. Debrecen 10 (1963), 10–13

    1963

  13. [13]

    Erdős and R

    P. Erdős and R. Rado, Combinatorial theorems on classifications of subsets of a given set, Proc. London Math. Soc.(3) 2 (1952), 417–439

    1952

  14. [14]

    Erdős and G

    P. Erdős and G. Szekeres, A combinatorial problem in geometry,Compos. Math.2 (1935), 463–470

    1935

  15. [15]

    C. Fan, X. Hu, Q. Lin and X. Lu, New bounds of two hypergraph Ramsey problems, arXiv preprintarXiv:2410.22019, 2024

    work page arXiv 2024

  16. [16]

    Fiz Pontiveros, S

    G. Fiz Pontiveros, S. Griffiths and R. Morris, The triangle-free process and the Ramsey numberR(3,k),Mem. Amer. Math. Soc.263 (2020), no. 1274, 125 pp. 7

    2020

  17. [17]

    R. L. Graham, B. L. Rothschild and J. H. Spencer, Ramsey Theory, 2nd edn.Wiley Interscience Series in Discrete Mathematics and Optimization(Wiley, New York, 1990)

    1990

  18. [18]

    Gupta, N

    P. Gupta, N. Ndiaye, S. Norin, and L. Wei, Optimizing the CGMS upper bound on Ramsey numbers, arXiv preprintarXiv:2407.19026, 2024

    work page arXiv 2024

  19. [19]

    Hefty, P

    Z. Hefty, P. Horn, D. King and F. Pfender, ImprovingR(3,k )in just two bites, arXiv preprintarXiv:2510.19718, 2025

    work page arXiv 2025

  20. [20]

    X. Hu, Q. Lin, X. Lu and G. Wang, Phase transitions of the Erdős-Gyárfás function, arXiv preprintarXiv:2504.05647, 2025

    work page arXiv 2025

  21. [21]

    Hunter, A

    Z. Hunter, A. Milojević and B. Sudakov, Gaussian random graphs and Ramsey numbers, arXiv preprintarXiv:2512.17718, 2025

    work page arXiv 2025

  22. [22]

    J. H. Kim, The Ramsey numberR(3,t )has order of magnitudet2/logt ,Random Struct. Algorithms.7 (1995), 173–207

    1995

  23. [23]

    Y. Li, C. C. Rousseau and W. Zang, An upper bound for Ramsey numbers,Appl. Math. Lett.17 (2004), 663–665

    2004

  24. [24]

    J. Ma, W. Shen and S. Xie, An exponential improvement for Ramsey lower bounds, arXiv preprintarXiv:2507.12926, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  25. [25]

    Mattheus and J

    S. Mattheus and J. Verstraëte, The asymptotics ofr(4,t ).Ann. of Math.199 (2024), 919–941

    2024

  26. [26]

    Mubayi and A

    D. Mubayi and A. Suk, New lower bounds for hypergraph Ramsey numbers,Bull. London Math. Soc.50 (2018), 189–201

    2018

  27. [27]

    Mubayi and A

    D. Mubayi and A. Suk, Off-diagonal hypergraph Ramsey numbers,J. Combin. Theory Ser. B125 (2017), 168–177

    2017

  28. [28]

    F. P. Ramsey, On a problem of formal logic,Proc. Lond. Math. Soc.30 (1930), 264–286

    1930

  29. [29]

    J. B. Shearer, A note on the independence number of triangle-free graphs.Discrete Math. 46 (1983), 83–87

    1983

  30. [30]

    Spencer, Asymptotic lower bounds for Ramsey functions,Discrete Math.20 (1977), 69–76

    J. Spencer, Asymptotic lower bounds for Ramsey functions,Discrete Math.20 (1977), 69–76. 8

    1977