pith. sign in

arxiv: 2605.31546 · v1 · pith:DZVMDN57new · submitted 2026-05-29 · 🧮 math.CO

Ramsey-Tur\'{a}n theory for partially-ordered sets

Gyula O.H. Katona , Yaping Mao This is my paper

Pith reviewed 2026-06-28 21:55 UTC · model grok-4.3

classification 🧮 math.CO
keywords poset Ramsey-Turán numberBoolean latticeweak and strong variantschainnon-chain posetasymptotic growthextremal set theory
0
0 comments X

The pith

Weak and strong poset Ramsey-Turán numbers satisfy RT ≤ RT^sharp with equality for chains, and the strong version grows as Theta(n^t) for any non-chain poset in the Boolean lattice.

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

The paper defines weak and strong poset Ramsey-Turán numbers that measure the largest subfamily of the Boolean lattice without a copy of a given poset P while controlling the size of certain t-chains. It establishes that the weak version is always at most the strong version, and the two are identical precisely when P forms a chain. For the special case of a single t-chain forbidden, exact values are obtained, while for non-chain posets the strong number is shown to be on the order of n to the power t. These results supply both exact equalities for chains and matching upper and lower bounds that determine the asymptotic order in the Boolean lattice setting.

Core claim

For any poset P the inequality RT(B;n,P,l,t) ≤ RT^sharp(B;n,P,l,t) holds, with equality whenever P is a chain. In particular, when t equals 1 and P is the chain C_k both quantities equal (k-1)(l-1). For every non-chain poset P the strong number RT^sharp(B;n,P,l,t) is Theta(n^t) once l and t are fixed; the same Theta(n^t) growth is proved for the poset A_k under the condition that min{l-1,k-1} is at least 1.

What carries the argument

The weak poset Ramsey-Turán number RT and the strong poset Ramsey-Turán number RT^sharp, which bound the size of a subfamily of the Boolean lattice that avoids a copy of P while limiting t-chains to length at most l.

If this is right

  • When the forbidden poset is a chain the weak and strong numbers coincide for every choice of parameters.
  • For t equal to 1 the common value on a chain C_k is exactly (k-1)(l-1).
  • The strong number grows polynomially of degree t for every non-chain poset once l and t are fixed.
  • Both versions admit lower bounds of order 2 to the beta n times n to the minus beta over 2 when the height of P exceeds t and l(n) grows like M_n to the beta.

Where Pith is reading between the lines

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

  • The separation between weak and strong versions may produce qualitatively different growth rates when the host family is changed from the Boolean lattice to another graded poset.
  • The polynomial growth for non-chains suggests that the extremal function behaves more like a Turán-type density than an exponential Ramsey-type quantity once the forbidden structure is not a chain.
  • The exact linear formula for chains when t equals 1 supplies a benchmark that could be used to test whether similar closed-form expressions exist for other small-height posets.

Load-bearing premise

The newly introduced definitions of the weak and strong poset Ramsey-Turán numbers are the natural and correct extensions of the classical graph-theoretic notions to the poset setting, and the Boolean lattice family admits the stated comparisons and asymptotic statements.

What would settle it

A concrete counter-example would be a non-chain poset P together with fixed l and t such that the strong number RT^sharp(B;n,P,l,t) is not Theta(n^t), for instance by exhibiting a family whose size is o(n^t) or omega(n^t) while satisfying the forbidden-subposet condition.

read the original abstract

We introduce weak and strong poset Ramsey-Tur\'an numbers for $t$-chains in host poset families, focusing on the Boolean lattice family $\mathcal{B}=\{B_n:n\ge 1\}$. For any poset $P$, we show $\operatorname{RT}(\mathcal{B};n,P,l,t)\le \operatorname{RT}^{\sharp}(\mathcal{B};n,P,l,t)$, with equality when $P$ is a chain. In particular, for $t=1$, $\operatorname{RT}(\mathcal{B};n,C_k,l)=\operatorname{RT}^{\sharp}(\mathcal{B};n,C_k,l)=(k-1)(l-1)$. We also give universal upper bounds for both versions. For fixed $k,l,t$ with $\min\{l-1,k-1\}\ge 1$, we prove $\operatorname{RT}^{\sharp}(\mathcal{B};n,A_k,l,t)=\Theta(n^t)$. More generally, for every non-chain poset $P$, the strong number is $\Theta(n^t)$ for fixed $l,t$. Finally, if $h(P)=r>t$ and $l(n)=\lfloor M_n^\beta\rfloor$ with $0<\beta\le \alpha<1$, then both weak and strong versions admit lower bounds of order $\Omega\!\left(2^{\beta n}n^{-\beta/2}\right)$.

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 manuscript introduces the weak and strong poset Ramsey-Turán numbers RT(B;n,P,l,t) and RT^sharp(B;n,P,l,t) for t-chains in the Boolean lattice family B={B_n:n≥1}. It establishes RT(B;n,P,l,t) ≤ RT^sharp(B;n,P,l,t) for any poset P, with equality when P is a chain; gives the exact value (k-1)(l-1) for both when t=1 and P=C_k; proves RT^sharp(B;n,A_k,l,t)=Θ(n^t) for fixed k,l,t with min{l-1,k-1}≥1 and more generally that the strong number is Θ(n^t) for every non-chain P; and derives Ω(2^{βn} n^{-β/2}) lower bounds for both versions when h(P)=r>t and l(n)=⌊M_n^β⌋ with 0<β≤α<1.

Significance. If the stated inequalities, exact constants, and Θ(n^t) asymptotics hold, the work supplies a direct combinatorial extension of classical Ramsey-Turán theory to posets, with explicit parameter-free values and growth rates that could serve as a basis for further extremal results on Boolean lattices. The equality case for chains and the distinction between weak and strong variants are clearly articulated strengths.

minor comments (3)
  1. Abstract: the family notation B={B_n:n≥1} is introduced without an explicit reminder that B_n denotes the power set of [n] ordered by inclusion; a parenthetical clarification would aid readers new to the poset setting.
  2. The distinction between the weak and strong numbers is central; ensure the introductory section supplies a side-by-side comparison of their definitions before the inequality is stated.
  3. The lower-bound construction when l grows with β should include a brief indication of the probabilistic or counting method used, even if details appear later.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main results, and recommendation for minor revision. We are pleased that the work is viewed as providing a direct extension of classical Ramsey-Turán theory to posets with explicit values and growth rates.

Circularity Check

0 steps flagged

No circularity; all stated results are direct theorems from newly introduced definitions

full rationale

The paper defines new weak and strong poset Ramsey-Turán numbers RT and RT^sharp as extensions of graph-theoretic notions to posets on the Boolean lattice. It then proves the inequality RT ≤ RT^sharp (with equality on chains), the exact value (k-1)(l-1) for t=1 on chains, and the Θ(n^t) asymptotics for antichains and non-chains via combinatorial arguments. No step reduces by construction to a fitted parameter, self-citation, or redefinition of the input; the derivations are self-contained and externally falsifiable through explicit constructions and bounds.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Review performed on abstract only; ledger therefore limited to quantities explicitly named. The central claims rest on the new definitions of RT and RT^sharp and on standard facts about the Boolean lattice.

axioms (1)
  • domain assumption The Boolean lattice B_n is the power set of [n] ordered by inclusion, and standard subposet containment applies.
    Invoked throughout the abstract as the host family.
invented entities (2)
  • weak poset Ramsey-Turán number RT(B;n,P,l,t) no independent evidence
    purpose: Quantifies the maximum size of a subfamily of B_n avoiding P while containing at least l t-chains.
    Newly introduced in the paper; no independent evidence supplied.
  • strong poset Ramsey-Turán number RT^sharp(B;n,P,l,t) no independent evidence
    purpose: A variant of the weak number, shown to be at least as large.
    Newly introduced in the paper; no independent evidence supplied.

pith-pipeline@v0.9.1-grok · 5779 in / 1569 out tokens · 26360 ms · 2026-06-28T21:55:26.663045+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

41 extracted references · 2 canonical work pages

  1. [1]

    Axenovich and S

    M. Axenovich and S. Walzer, Boolean lattices: Ramsey properties and embeddings,Order34 (2017), 287–298

    2017

  2. [2]

    Axenovich and C

    M. Axenovich and C. Winter, Poset Ramsey numbers: large Boolean lattice versus a fixed poset,Combin. Probab. Comput.32 (2023), 638–653

    2023

  3. [3]

    Axenovich and C

    M. Axenovich and C. Winter, Poset Ramsey numberR(P, Q n). II.N-shaped poset,Order41 (2024), 401–418

    2024

  4. [4]

    Balogh, H

    J. Balogh, H. Liu, and M. Sharifzadeh, On two problems in Ramsey–Tur´ an theory,SIAM J. Discrete Math.31 (2017), 1848–1866

    2017

  5. [5]

    Balogh, R

    J. Balogh, R. R. Martin, D. T. Nagy, and B. Patk´ os, On generalized Tur´ an results in height two posets,SIAM J. Discrete Math.36(2) (2022), 1483–1495

    2022

  6. [6]

    Balogh, A

    J. Balogh, A. McDowell, T. Molla, and R. Mycroft, Triangle-tilings in graphs without large independent sets,Combin. Probab. Comput.27 (2018), 449–474

    2018

  7. [7]

    Bohman and F

    T. Bohman and F. Peng, A construction for Boolean cube Ramsey numbers,Order40 (2023), 327–333

    2023

  8. [8]

    Bollob´ as and P

    B. Bollob´ as and P. Erd˝ os, On a Ramsey–Tur´ an type problem,J. Combin. Theory Ser. B21 (1976), 166–168

    1976

  9. [9]

    Buci´ c, M

    M. Buci´ c, M. Christoph, J. Kim, H. Lee, and V. Sivashankar, On a Ramsey–Tur´ an variant of Roth’s theorem, arXiv:2507.22831 [math.CO], 2025

    work page arXiv 2025

  10. [10]

    Cox and D

    C. Cox and D. Stolee, Ramsey numbers for partially ordered sets,Order35 (2018), 557–579

    2018

  11. [11]

    R. P. Dilworth, A decomposition theorem for partially ordered sets,Ann. of Math.51 (1950), 161–166

    1950

  12. [12]

    Duffus, H

    D. Duffus, H. A. Kierstead, and W. T. Trotter, Fibres and ordered set coloring,J. Combin. Theory Ser. A58 (1991), 158–164

    1991

  13. [13]

    Erd˝ os and V

    P. Erd˝ os and V. T. S´ os, Some remarks on Ramsey’s and Tur´ an’s theorem, in:Combina- torial Theory and Its Applications, II(Proc. Colloq., Balatonf¨ ured, 1969), North-Holland, Amsterdam, 1970, pp. 395–404

    1969

  14. [14]

    Falgas-Ravry, K

    V. Falgas-Ravry, K. Markstr¨ om, A. Treglown, and Y. Zhao, Existence thresholds and Ramsey properties of random posets,Rand. Struct. Algor.57 (2020), 1097–1133. 20

    2020

  15. [15]

    R. L. Graham, B. L. Rothschild, and J. H. Spencer,Ramsey Theory, 2nd ed., John Wiley & Sons, New York, 1990

    1990

  16. [16]

    Gerbner and B

    D. Gerbner and B. Patk´ os,Extremal Finite Set Theory, CRC Press, Boca Raton, 2019

    2019

  17. [17]

    C. M. L¨ uders and C. Reiher, The Ramsey–Tur´ an problem for cliques,Israel J. Math.230 (2019), 613–652

    2019

  18. [18]

    Motwani and P

    R. Motwani and P. Raghavan,Randomized Algorithms, Cambridge University Press, Cam- bridge, 1995

    1995

  19. [19]

    Gr´ osz, A

    D. Gr´ osz, A. Methuku, and C. Tompkins, Ramsey numbers of Boolean lattices,Bull. Lond. Math. Soc.55 (2023), 914–932

    2023

  20. [20]

    Gerbner, B

    D. Gerbner, B. Keszegh, and B. Patk´ os, Generalized forbidden subposet problems,Order37 (2020), 389–410

    2020

  21. [21]

    Gerbner, A

    D. Gerbner, A. Methuku, D. T. Nagy, B. Patk´ os, and M. Vizer, On the number of containments inP-free families,Graphs Combin.35(6) (2019), 1519–1540

    2019

  22. [22]

    Gerbner and B

    D. Gerbner and B. Patk´ os,l-chain profile vectors,SIAM J. Discrete Math.22(1) (2008), 185–193

    2008

  23. [23]

    J. R. Griggs, J. Stahl, and W. T. Trotter, Jr., A Sperner theorem on unrelated chains of subsets,J. Combin. Theory Ser. A36 (1984), 124–127

    1984

  24. [24]

    D. S. Gunderson, V. R¨ odl, and A. Sidorenko, Extremal problems for sets forming Boolean algebras and complete partite hypergraphs,J. Combin. Theory Ser. A88 (1999), 342–367

    1999

  25. [25]

    Johnston, L

    T. Johnston, L. Lu, and K. G. Milans, Boolean algebras and Lubell functions,J. Combin. Theory Ser. A136 (2015), 174–183

    2015

  26. [26]

    G. O. H. Katona, Two applications (for search theory and truth functions) of Sperner type theorems,Period. Math. Hungar.3 (1973), 19–26

    1973

  27. [27]

    G. O. H. Katona, Y. Mao, K. Ozeki, and Z. Wang, Ramsey numbers for partially ordered sets, arXiv:2512.14638 [math.CO], 2025

    work page arXiv 2025

  28. [28]

    H. A. Kierstead and W. T. Trotter, A Ramsey theoretic problem for finite ordered sets, Discrete Math.63 (1987), 217–223

    1987

  29. [29]

    Kleitman and G

    D. Kleitman and G. Markowsky, On Dedekind’s problem: the number of isotone Boolean functions. II,Trans. Amer. Math. Soc.213 (1975), 373–390. 21

    1975

  30. [30]

    Lu and J

    L. Lu and J. C. Thompson, Poset Ramsey numbers for Boolean lattices,Order39 (2022), 171–185

    2022

  31. [31]

    Luczak, J

    T. Luczak, J. Polcyn, and C. Reiher, On the Ramsey–Tur´ an density of triangles,Combina- torica42 (2022), 115–136

    2022

  32. [32]

    G. L. McColm, A Ramseyian theorem on products of trees,J. Combin. Theory Ser. A57 (1991), 68–75

    1991

  33. [33]

    Neˇ setˇ ril and V

    J. Neˇ setˇ ril and V. R¨ odl, Combinatorial partitions of finite posets and lattices–Ramsey lattices, Algebra Univers.19 (1984), 106–119

    1984

  34. [34]

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

    1930

  35. [35]

    Rosta, Ramsey theory applications,Electron

    V. Rosta, Ramsey theory applications,Electron. J. Combin., Dynamic Surveys DS13 (2004), 43 pp

    2004

  36. [36]

    Simonovits and V

    M. Simonovits and V. T. S´ os, Ramsey–Tur´ an theory,Discrete Math.229 (2001), 293–340

    2001

  37. [37]

    W. T. Trotter, Ramsey theory and partially ordered sets, in:Contemporary Trends in Discrete Mathematics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 49, Amer. Math. Soc., Providence, RI, 1999, pp. 337–347

    1999

  38. [38]

    Tur´ an, Eine Extremalaufgabe aus der Graphentheorie,Mat

    P. Tur´ an, Eine Extremalaufgabe aus der Graphentheorie,Mat. Fiz. Lapok48 (1941), 436–452

    1941

  39. [39]

    Walzer,Ramsey Variant of the2-Dimension of Posets, Master’s thesis, Karlsruhe Institute of Technology, 2015

    S. Walzer,Ramsey Variant of the2-Dimension of Posets, Master’s thesis, Karlsruhe Institute of Technology, 2015

    2015

  40. [40]

    Winter, Poset Ramsey numberR(P, Q n)

    C. Winter, Poset Ramsey numberR(P, Q n). I. Complete multipartite posets,Order41 (2024), 391–399

    2024

  41. [41]

    Winter, Poset Ramsey numberR(P, Q n)

    C. Winter, Poset Ramsey numberR(P, Q n). III. Chain compositions and antichains,Discrete Math.347 (2024), 114031. 22

    2024