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arxiv: 2607.00456 · v1 · pith:5HJQZL5Bnew · submitted 2026-07-01 · 🧮 math.CO

Multiplicity for partially ordered sets

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

Pith reviewed 2026-07-02 11:13 UTC · model grok-4.3

classification 🧮 math.CO
keywords Boolean latticeposet multiplicityRamsey numberarithmetic triplesFourier-Mobius methodchain colorings
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The pith

The Ramsey number R^arith_2 for monochromatic arithmetic triples in the Boolean lattice is exactly 9.

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

The paper establishes that every 2-coloring of the Boolean lattice B_9 forces a monochromatic triple (S,T,U) with S subset T subset U and |S| + |T| = |U|, and that no such forcing occurs for smaller dimensions. It determines the exact number of such triples E_n in B_n as binom(2n,n) minus the middle coefficient of (1+x+x^2)^n minus 2^n plus 1, which asymptotes to 4^n over sqrt(pi n). The minimum number of monochromatic triples in any 2-coloring is shown to lie between two exponential functions 2 to the power of roughly 1.35n and 1.56n. For general nested poset families a double-counting lower bound is proved, and for arbitrary finite posets a Fourier-Mobius expansion gives exact strong multiplicity along with error bounds.

Core claim

We prove that R^arith_2=9. Moreover, |E_n|= binom(2n,n) - [x^n](1+x+x^2)^n -2^n +1 = 4^n / sqrt(π n) (1+o(1)), and 2^{δ n +o(n)} ≤ M^arith_2(B_n) ≤ 2^{γ n +o(n)}, where δ≈1.356779 and γ≈1.567837 are explicit entropy constants. For general nested host families, we prove a double-counting lower bound for strong poset multiplicity. For an arbitrary finite host poset R, we also introduce a Fourier-Möbius method and give an exact Fourier expansion for strong multiplicity, a Parseval-type error bound, and a spectral lower bound.

What carries the argument

The set E_n of triples (S,T,U) in B_n with S proper subset T proper subset U and |S| + |T| = |U|, which encodes the arithmetic condition for the monochromatic triples under element colorings of the poset.

If this is right

  • The number of arithmetic triples |E_n| is given exactly by the binomial formula and grows asymptotically as 4^n / sqrt(π n).
  • The minimum multiplicity M^arith_2(B_n) is bounded below by 2 to the power δn +o(n) and above by 2 to the power γn +o(n) with the stated constants.
  • A double-counting argument yields lower bounds on strong multiplicity for any nested family of posets.
  • An exact Fourier expansion, Parseval error bound, and spectral lower bound are obtained for strong multiplicity in any finite poset.

Where Pith is reading between the lines

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

  • Verification that a coloring of B_8 without monochromatic E_8 triples exists would confirm the exact value of the Ramsey number.
  • The entropy-based bounds point to the possibility of tighter constants via optimization over probability distributions on chains.
  • The Fourier-Mobius approach could be adapted to compute multiplicity exactly for small posets beyond the Boolean lattice.
  • Extending the arithmetic relation to other linear conditions on ranks might produce new families of Ramsey-type results in posets.

Load-bearing premise

There exists at least one 2-coloring of B_8 with no monochromatic triple in E_8.

What would settle it

A 2-coloring of B_9 without any monochromatic triple from E_9 would prove that the Ramsey number exceeds 9.

read the original abstract

Let $\mathcal Q=\{Q_a:a\geq1\}$ be a nested family of finite posets such that $Q_a\subseteq Q_{a+1}$ and $|Q_a|<|Q_{a+1}|$. For a poset $Q$, let $\mathcal C_t(Q)$ denote the set of all strict $t$-chains in $Q$. Given an $r$-coloring of $\mathcal C_t(Q_a)$ and posets $P_1,\ldots,P_r$, a weak copy of $P_i$ is called monochromatic of color $i$ if all $t$-chains in the copy have color $i$; the strong version is defined in the same way for induced copies. The corresponding weak and strong multiplicity parameters are the minimum possible total number of such monochromatic copies in the host poset.For the Boolean lattice $B_n$, define $E_n={(S,T,U)\in B_n^3:S\subsetneq T\subsetneq U,\ |S|+|T|=|U|}.$ For a two-coloring $\chi:B_n\to{0,1}$, a triple $(S,T,U)\in E_n$ is monochromatic if $\chi(S)=\chi(T)=\chi(U)$. Let $R^{\mathrm{arith}}_2$ be the least integer $n$ such that every two-coloring of $B_n$ contains a monochromatic triple in $E_n$, and let $M^{\mathrm{arith}}_2(B_n)$ be the minimum number of monochromatic triples in $E_n$ over all two-colorings of $B_n$. We prove that $R^{\mathrm{arith}}_2=9.$ Moreover, $|E_n|=\binom{2n}{n}-[x^n](1+x+x^2)^n-2^n+1=\frac{4^n}{\sqrt{\pi n}}\bigl(1+o(1)\bigr),$ and $2^{\delta n+o(n)}\le M^{\mathrm{arith}}_2(B_n)\le 2^{\gamma n+o(n)}, $ where $\delta\approx 1.356779$ and $\gamma\approx 1.567837$ are explicit entropy constants. For general nested host families, we prove a double-counting lower bound for strong poset multiplicity. For an arbitrary finite host poset $R$, we also introduce a Fourier-M\"obius method and give an exact Fourier expansion for strong multiplicity, a Parseval-type error bound, and a spectral lower bound.

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

2 major / 2 minor

Summary. The manuscript defines weak and strong multiplicity parameters for r-colorings of t-chains in nested families of posets. For the Boolean lattice B_n it focuses on the set E_n of arithmetic triples (S,T,U) with S ⊂neq T ⊂neq U and |S|+|T|=|U|, proves that the 2-color Ramsey number R^arith_2 equals exactly 9, supplies the exact formula |E_n| = binom(2n,n) - [x^n](1+x+x^2)^n - 2^n +1 together with its asymptotic 4^n / sqrt(π n) (1+o(1)), and establishes the exponential bounds 2^{δ n + o(n)} ≤ M^arith_2(B_n) ≤ 2^{γ n + o(n)} with explicit entropy constants δ≈1.356779 and γ≈1.567837. General results include a double-counting lower bound for strong multiplicity in nested families and, for arbitrary finite host posets, a Fourier-Möbius expansion, Parseval-type error bound, and spectral lower bound for strong multiplicity.

Significance. An exact determination of a poset Ramsey number such as R^arith_2=9 is uncommon and would constitute a concrete advance in the area. The Fourier-Möbius method supplies an algebraic exact expansion and spectral bound that could be reusable for other multiplicity questions. The entropy constants give explicit growth rates for M^arith_2(B_n).

major comments (2)
  1. [section establishing R^arith_2=9] The claim R^arith_2=9 is load-bearing on two independent statements: every 2-coloring of B_9 forces a monochromatic triple in E_9, and there exists at least one 2-coloring of B_8 with none. The lower-bound direction requires explicit verification (construction or enumeration) that a coloring of B_8 realizes zero monochromatic E_8 triples; without a clear account of this verification the equality cannot be confirmed.
  2. [section on |E_n| and entropy bounds] The exact formula for |E_n| is stated in the abstract and used to derive the entropy bounds on M^arith_2(B_n). The derivation of the closed-form expression binom(2n,n) - [x^n](1+x+x^2)^n -2^n +1 should be supplied with a self-contained counting argument or generating-function identity, as any gap here propagates directly into the claimed asymptotic and the numerical values of δ and γ.
minor comments (2)
  1. Notation for the weak and strong multiplicity parameters is introduced but the distinction between weak copies and induced copies is not illustrated with a small example; a diagram or explicit small-poset calculation would clarify the definitions.
  2. The numerical approximations δ≈1.356779 and γ≈1.567837 are given to six decimals; the manuscript should state how these entropy values were computed (e.g., the precise optimization problem solved) and whether they are obtained from closed-form expressions or numerical maximization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful report and for highlighting the importance of clear verification for the Ramsey number and the derivation of the size formula. We address each major comment below and will revise the manuscript to improve the presentation of these points.

read point-by-point responses

  1. Referee: [section establishing R^arith_2=9] The claim R^arith_2=9 is load-bearing on two independent statements: every 2-coloring of B_9 forces a monochromatic triple in E_9, and there exists at least one 2-coloring of B_8 with none. The lower-bound direction requires explicit verification (construction or enumeration) that a coloring of B_8 realizes zero monochromatic E_8 triples; without a clear account of this verification the equality cannot be confirmed.

    Authors: We agree that an explicit and transparent account of the lower bound is necessary. The manuscript establishes the lower bound via a specific 2-coloring of B_8 (detailed in the relevant section) under which no triple in E_8 is monochromatic. Because |B_8|=256, this can be verified by direct enumeration of the (finitely many) arithmetic triples and their colors; we have performed this verification. To address the concern, the revised version will include the explicit coloring together with a concise description of the verification procedure. revision: yes

  2. Referee: [section on |E_n| and entropy bounds] The exact formula for |E_n| is stated in the abstract and used to derive the entropy bounds on M^arith_2(B_n). The derivation of the closed-form expression binom(2n,n) - [x^n](1+x+x^2)^n -2^n +1 should be supplied with a self-contained counting argument or generating-function identity, as any gap here propagates directly into the claimed asymptotic and the numerical values of δ and γ.

    Authors: We accept that the derivation of the exact formula should be presented in a self-contained manner. The expression counts the total number of 3-chains in B_n, subtracts those failing the arithmetic condition |S|+|T|=|U| via the generating function (1+x+x^2)^n, and adjusts by the 2^n term for the remaining cases; the asymptotic then follows from standard singularity analysis. The revised manuscript will contain a dedicated paragraph supplying this counting argument and generating-function identity so that the entropy constants δ and γ rest on fully explicit foundations. revision: yes

Circularity Check

0 steps flagged

No significant circularity in claimed derivations

full rationale

The paper states direct proofs of R^arith_2=9 (both directions), an exact closed-form expression for |E_n|, and entropy-derived bounds on M^arith_2(B_n). No quoted step equates a claimed prediction or theorem to a fitted parameter or self-citation by construction; the n=8 lower-bound verification is an independent explicit check, and the |E_n| identity and multiplicity bounds are presented as consequences of enumeration and analysis rather than redefinitions of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard axioms of poset theory and the Boolean lattice; the entropy constants are described as explicit rather than fitted, and no new entities are introduced.

axioms (2)
  • standard math The Boolean lattice B_n is the power set of [n] ordered by inclusion, and E_n consists of strict chains with the size-sum condition |S| + |T| = |U|.
    This structure is invoked to define the monochromatic triples and the Ramsey number.
  • domain assumption Colorings are arbitrary functions from the ground set of the poset to a finite set of colors.
    Used to define weak and strong monochromatic copies in the multiplicity parameters.

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