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Schrijver Number Quasi-Tensorization and Multicolor Ramsey Bounds via Robust OR Polynomials

T0 review · 1 major / 6 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read A single polynomial recipe turns one-coordinate PSD certificates into OR certificates, yielding quasi-tensorization of the Schrijver number and tighter multicolor Ramsey bounds.

desk verdict Clean methods paper: a reusable nonnegative OR polynomial + compressors gives the first quasi-tensorization of Schrijver numbers and a real r-exponent improvement on the BBC+26 multicolor Ramsey bound. read the letter →

arxiv 2607.25023 v1 pith:7J74TJQ4 submitted 2026-07-27 math.CO cs.DM

Ijay Narang , Yukai Tang This is my paper · ORCID
classification math.COcs.DM MSC 05C5505D1090C2205C69
keywords Schrijvernumberstrongproductacute-freefamiliesmulticolorRamseynumbersORpolynomialspositivesemidefinitekernelsgeometriclemmabookalgorithm
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

Many extremal problems ask that at least one of several coordinates satisfies a bad event. One-coordinate versions often have clean positive-semidefinite certificates, but those certificates do not automatically compose under an OR. This paper builds a robust OR polynomial with nonnegative coefficients that detects a bad coordinate while preserving positive-semidefiniteness after composition with compressors. Applied to the Schrijver number, the method gives a general quasi-tensorization bound under strong products, which controls the size of r-way acute-free families of vectors by a quasipolynomial in n. The same tool improves the geometric input to an existing book algorithm for multicolor Ramsey numbers, shaving the r-dependence in the exponential error term. A sympathetic reader cares because the method turns a structural obstruction—non-multiplicativity of a tight SDP bound—into a usable approximate product rule and feeds a better local lemma into a known global algorithm.

What carries the argument

The robust OR polynomial Qr(t) = ∏(1+ti) − ∏(1+(1−1/r)ti), whose nonnegative coefficients detect a coordinate near −1 and, by Schur products, keep composed kernels positive semidefinite.

What would settle it

Exhibit maps σ1,…,σr for which some nonnegative multi-index has negative inner-product moment, or compute ϑ′ of an explicit strong product of small Schrijver graphs and check whether it exceeds the claimed product-of-powers bound.

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Extended reading notes

Core claim

There is a robust OR polynomial, built from elementary symmetric sums with nonnegative coefficients, that becomes strictly negative whenever any input lies near −1 while remaining controlled when all inputs stay at least −1. Composed with sign or Bessel compressors that map one-coordinate PSD kernels into that range, it produces dual feasible kernels for strong products and nonnegative-expectation test functions for vector-valued maps. Consequently the Schrijver number of a strong product is at most a product of the factors raised to C log r log ϑ′, and the multicolor Ramsey number satisfies Rr(k) ≤ exp(−Ω(k/(r9 (log r)6))) r^{rk} for large enough k.

Load-bearing premise

The Ramsey improvement rests on an inherited fact that every monomial moment of the inner-product coordinates is nonnegative, so any test function with nonnegative Taylor coefficients has nonnegative expectation.

Editorial extensions

If this is right

  • r-way acute-free families on the sphere or hypercube have size at most (2n)^{O(r log r log(2n))}, closing most of the gap from the trivial (2n)^r lower bound to a quasipolynomial.
  • Schrijver numbers of strong products admit a uniform quasi-multiplicative upper bound whenever each factor has ϑ′ ≥ 2.
  • The geometric lemma feeding the multicolor book algorithm improves from Cr = Θ(r^{3/2}) to Cr = O(r log r), yielding Rr(k) ≤ exp(−Ω(k/(r^9 (log r)^6))) r^{rk}.
  • The same compressor-plus-OR template applies to any OR-type extremal problem that already possesses one-coordinate PSD or moment certificates.

Reading between the lines

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

  • If the degree of the sign compressor can be reduced below log ϑ′, the quasi-tensorization exponent could drop from quasipolynomial toward the pure product form conjectured for two factors.
  • The same OR polynomial may give approximate multiplicativity for other non-multiplicative SDP hierarchy numbers that differ from Lovász theta only by nonnegativity constraints.
  • Numerical checks that M2(n) is n^{2+o(1)} would support attacking the two-factor Schrijver conjecture via a tighter Delsarte LP rather than further polynomial engineering.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 6 minor

Summary. The paper introduces a "robust OR polynomial" Q_r(t)=∏(1+t_i)−∏(1+(1−1/r)t_i), which has nonnegative coefficients and detects the event that some coordinate lies in [−1,−1+η], and uses it to compose one-coordinate PSD certificates across OR constraints via Schur products. Two applications are given. First (Theorem 1.1), a quasi-tensorization theorem for the Schrijver number: for graphs with ϑ′(G_i)≥2, ϑ′(⊠_i G_i) ≤ ∏_i ϑ′(G_i)^{C log r·log ϑ′(G_i)}, proved by building a dual-feasible kernel from the one-coordinate Schrijver dual solutions passed through a sinc-quadrature sign compressor (Lemma 2.2) and composed with Q_r. This yields quasipolynomial bounds M_r(n) ≤ (2n)^{C_0 r log r log(2n)} for r-way acute-free families (Corollary 1.2). Second (Lemma 1.3, Theorem 1.5), an improvement of the geometric lemma of Balister et al. (BBC+26), replacing C_r=Θ(r^{3/2}) by C_r=O(r log r) via a Bessel-function compressor with nonnegative Taylor coefficients and exp(O(log r)√(x+1)) growth; fed into the unchanged BBC+26 book algorithm, this gives R_r(k) ≤ exp(−ck/(r^9(log 2r)^6)) r^{rk} for k ≥ C_0 r^{14}(log 2r)^{12}, improving the previous exp(−Ω(k/r^{12}))r^{rk}.

Significance. If correct, the results are strong. The improvement of the multicolor Ramsey exponent from r^{12} (a JAMS breakthrough from 2024/26) to r^9(log r)^6 is substantial progress on a high-profile problem, and it is achieved by a clean modular substitution: the book algorithm is untouched, only the geometric input is improved. The Schrijver quasi-tensorization theorem appears to be the first general statement of its kind and the quasipolynomial bound on M_r(n) answers (up to the exponent) a problem circulated by the BBC+26 authors. Strengths worth naming: the proofs are complete and largely self-contained; the one external load-bearing input (moment positivity, BBC+26 Lemma 3.2) is properly isolated and used verbatim; the certificate construction is parameter-free in the sense that all constants are absolute and nothing is fitted; and the framework itself (compressor + nonnegative-coefficient OR polynomial + Schur products) is a reusable method likely to find further applications. I verified the main chains independently: the dual-certificate Lemma 2.1, the compressor bounds in Theorem 1.1 (including the 2×2-minor bound |Z_j(x,y)| ≤ 2λ_j−1 that requires ϑ′≥2), the expansion (8) giving K−

major comments (1)
  1. [§A.2, proof of Lemma 3.3 and Lemma A.3] Proof of Lemma 3.3 (Appendix A.2, p. 24) and Lemma A.3 hypothesis (p. 21): the verification of the hypothesis of Lemma A.3 contains wrong-direction inequalities, and as stated Lemma 3.3's hypotheses do not imply Lemma A.3's condition. Specifically: (i) Lemma A.3 assumes log(r/β_r) ≤ μ_r/(16 log(μ_r²/p_r)), but the natural quantity the proof actually uses is μ_r·log(1/δ_r)/16 with the logarithm in the numerator — note log(1/δ_r)=log(μ_r²/p_r); the subsequent displayed bounds log(r/β_r) ≤ μ_r/(16 log(1/δ_r)) and C_r√(λ_{0,r}+1) ≤ √2 μ_r/(8 log(1/δ_r)) both have log(1/δ_r) in the denominator, whereas direct computation from λ_{0,r}=(μ_r log(1/δ_r)/(8C_r))² gives C_r√(λ_{0,r}+1) ≤ √2 μ_r log(1/δ_r)/8, with the logarithm in the numerator. (ii) Consequently, in Lemma 3.3 the chain log(r/β_r) ≤ M_r log(2r) ≤ μ_r/(16 log(μ_r²/p_r)) has a false second inequality: with μ_r=AM_r and p_r≥1/(2r), log
minor comments (6)
  1. [§3, Fact 3.1 vs. proof of Lemma 3.3] Notation collision: the letter A denotes the absolute Bessel cutoff of Fact 3.1 (used throughout §3.1) and is reused as the large absolute constant in the proof of Lemma 3.3 (p. 23). Please rename one of them.
  2. [§2, proof of Theorem 1.1] In the proof of Theorem 1.1, the identification λ_i = ϑ′(G_i) (dual optimum equals primal) is used implicitly when concluding (2λ_i)^{D_i} ≤ ϑ′(G_i)^{C log r log ϑ′(G_i)}; a one-line reminder that strong duality holds for the Schrijver SDP [Sch79] would help readers. Similarly, the step |Z_j(x,y)| ≤ 2λ_j−1 from Z_j−J ⪰ 0 via the 2×2 principal minor deserves one displayed line, since it is where the hypothesis ϑ′(G_i) ≥ 2 enters.
  3. [§A.2, proof of Lemma A.3] In the proof of Lemma A.3, the displayed equality η_r^{2rt_r} ≥ exp(−μ_r r t_r/(2 log(1/δ_r))) = (μ_r²/p_r)^{−μ_r r t_r/2} has a false equality sign (exp(−a/log b) ≠ b^{−a}); the correct relation is '≥' via the numerator computation — see major comment. Please correct the display.
  4. [§4, Conjecture 4.1] Conjecture 4.1 is said to be 'supported by numerical evidence' (correct order of M_2(n) seems to be n^{2+o(1)}), but no data or description of the computation is given. Either include a brief description or soften the claim.
  5. [Abstract / §1] The abstract states the Ramsey bound with (log r)^6 while Theorem 1.5 uses (log(2r))^6; harmless, but please harmonize. Also, the absolute constants C, C_0, c in Theorems 1.1/1.5 are not estimated; a remark on whether the methods give explicit (if large) values would be useful.
  6. [§1, Theorem 1.1 statement] The hypothesis ϑ′(G_i) ≥ 2 of Theorem 1.1 excludes graphs with ϑ′<2 (e.g., complete graphs); a short remark that such factors can be handled separately (or absorbed into the product bound) would preempt a natural reader question.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: both main theorems are derived from dual SDP certificates, explicit compressors, and an external moment-positivity lemma, not from the target bounds.

full rationale

The paper’s two load-bearing claims are proved by constructing explicit PSD kernels / entire test functions and feeding them into standard dual or expectation arguments. Theorem 1.1 builds one-coordinate Schrijver dual kernels, compresses them with a proved sinc-quadrature rational sign approximant (Lemma 2.2), composes with the robust OR polynomial Q_r (nonnegative coefficients), and invokes Schur’s product theorem plus the elementary dual-certificate Lemma 2.1; the exponent is pure degree bookkeeping. Corollary 1.2 only instantiates that bound with the classical spherical-code kernel q(t)=n(t+t²). Lemma 1.3 likewise builds a Bessel compressor with verified Taylor nonnegativity and growth, composes with Q_r, and uses the external BBC+26 moment-positivity lemma (E∏Z_i^{a_i}≥0) to force mass on the good event; the improved Cr=O(r log r) is not assumed. Theorem 1.5 only re-traces the unchanged BBC+26 book algorithm with the new geometric parameters. There is no fitted parameter recycled as a prediction, no self-citation uniqueness theorem, and no definition of the target quantities in terms of themselves. The sole external load-bearing input is a refereed JAMS lemma used exactly as stated. Score 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 2 invented entities

The paper rests on standard SDP duality and kernel calculus, classical special-function bounds, and one external combinatorial lemma from BBC+26. Absolute constants are existential, not data-fitted. The robust OR polynomial and compressors are invented tools with explicit formulas, not free physical entities.

free parameters (3)
  • Absolute constant C (quasi-tensorization exponent)
    Exists from degrees of Fi,Hi and the even power L; not numerically optimized or fitted to data.
  • Absolute constant C0 in Mr(n) and Ramsey k-threshold
    Propagates from C and from book-algorithm constants; existential only.
  • Bessel cutoff A and growth constant C1 = A≥1 existential
    A is chosen so |J0(2√a)|≤e^{-3} for a≥A (Fact 3.1); C1 absorbs 4m√A with m=⌈(1/6)log(8er)⌉. Standard analytic constants, not fit to combinatorial data.
assumptions (5)
  • standard math Strong duality for the Schrijver SDP (cited Schrijver 1979)
    Used to pass from dual kernels to ϑ' upper bounds (Lemma 2.1, §2).
  • standard math Schur product theorem: entrywise products of PSD matrices remain PSD
    Preserves PSD-ness when composing nonnegative-coefficient polynomials of kernels (§1.1, proof of Thm 1.1).
  • domain assumption Moment positivity: E[∏ Zi^{ai}]≥0 for Zi=⟨σi(U),σi(U')⟩ (BBC+26 Lemma 3.2)
    Black-box input that lets any nonnegative-Taylor entire function serve as a test function for the geometric lemma.
  • standard math Asymptotic decay |J0(x)|→0 and integral bound I0(x)≤e^x for modified Bessel I0
    Fact 3.1 and growth of T(x) in Lemma 3.2 (Watson treatise).
  • domain assumption Book-algorithm analysis of BBC+26 continues to hold when (Cr,βr) are replaced by the improved parameters
    Lemma 3.3 / Appendix A.2 re-traces their lemmas with tracked Cr,βr; no new algorithmic idea.
invented entities (2)
  • Robust OR polynomial Qr(t)=∏(1+ti)−∏(1+(1−1/r)ti) independent evidence
    purpose: Detect any coordinate in [−1,−1+η] while keeping nonnegative coefficients for PSD composition
    Defined in (5); Lemma 1.6 proves the detection property. Explicit elementary formula, independently checkable.
  • Sign/Bessel compressors Ti and T independent evidence
    purpose: Map one-coordinate certificates into the working range of Qr with controlled growth
    Rational sinc-quadrature approximants (Lemma 2.2) and Ψ(Ax)=B(Ax)^{2m} (18). Constructive, not postulated particles.

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Pith. "Pith review of Schrijver Number Quasi-Tensorization and Multicolor Ramsey Bounds via Robust OR Polynomials." pith.science (2026). https://pith.science/paper/7J74TJQ4

@misc{pith2026260725023,
  author       = {Pith},
  title        = {Pith review of: Schrijver Number Quasi-Tensorization and Multicolor Ramsey Bounds via Robust OR Polynomials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7J74TJQ4}},
  note         = {Machine review of arXiv:2607.25023}
}
abstract

We introduce a robust OR polynomial framework for composing positive semidefinite certificates across OR constraints. We demonstrate the power of this method in two applications. The first is on acute-free families. A set $\mathcal F=\{(x_i^{(1)},\ldots,x_i^{(r)})\}_{i=1}^M \subseteq (S^{n-1})^r$ is $r$-way acute-free if, for every $i\neq j$, there is a coordinate $t\in[r]$ such that $\langle x_i^{(t)},x_j^{(t)}\rangle\leq 0$. We write $M_r(n)$ for the maximum size of such a set, and $M_r^{\pm}(n)$ for the hypercube restriction. On the hypercube, $r$-way acute-free sets are independent sets for some strong power graph $G_n^{\boxtimes r}$. The Lov\'asz theta number $\vartheta(G_n)$ is multiplicative but exponentially loose, whereas the Schrijver number $\vartheta'(G_n)$ gives the correct order, but is not multiplicative. We bypass this obstruction by proving a general quasi-tensorization result for the Schrijver number. That is, for every collection of graphs $G_1,\ldots, G_r$ satisfying $\vartheta'(G_i)\geq 2$, there is an absolute constant $C$ such that $\vartheta'(G_1\boxtimes \cdots \boxtimes G_r) \leq \prod_{i=1}^r \vartheta'(G_i)^{C\log r \log \vartheta'(G_i)}$. Applying this result gives that $M_r^{\pm}(n) \le M_r(n) \le (2n)^{C_0 r\log r\log(2n)}$ for some absolute constant $C_0$. The second application is on multicolor Ramsey numbers. The $r$-color Ramsey number $R_r(k)$ is the minimum $n$ such that every $r$-coloring of the edges of the complete graph on $n$ vertices contains a monochromatic copy of $K_k$. In a breakthrough result, Balister et al. [arXiv:2410.17197] showed that $R_r(k)\le \exp(-\Omega(k/r^{12}))r^{rk}$ via a geometric lemma. By improving the $r$ dependency in their geometric lemma via the OR polynomial framework, we prove that $R_r(k)\le \exp(-\Omega(k/(r^9(\log r)^6)))r^{rk}$.

Figures

Figures reproduced from arXiv: 2607.25023 by the authors.

Figure 1
Figure 1. Diagrammatic Workflow of the robust OR polynomial framework. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

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Cited by 1 Pith paper

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

  1. New upper bound for multicolor Ramsey numbers

    math.CO 2026-08 accept novelty 8.0 of 10

    For r≥2 and k≥K r^2 log^6(2r), the paper proves R_r(k) ≤ exp(-c k/(r^2 log^4(2r))) r^{rk}, improving the previously known multicolor Ramsey upper bound.

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