Density Frankl-R\"{o}dl on the Sphere

Shilun Li; Venkatesan Guruswami

arxiv: 2505.09839 · v4 · submitted 2025-05-14 · 🧮 math.PR

Density Frankl-R\"{o}dl on the Sphere

Venkatesan Guruswami , Shilun Li This is my paper

Pith reviewed 2026-05-22 14:33 UTC · model grok-4.3

classification 🧮 math.PR

keywords Frankl-Rödl theoremdensity versionsphereinner productmeasurable setinductive configurationsphere Ramseyavoidance problem

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The pith

Large measurable sets on the sphere must contain pairs with any prescribed inner product.

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

This paper proves a density version of the Frankl-Rödl theorem on the sphere. For any fixed inner product, if a measurable subset A has large enough measure then the probability that two random vectors with that inner product both lie in A is bounded below by a positive constant. The argument extends to density statements for broader configurations that prescribe multiple pairwise inner products. The framework covers a class of inductive configurations that includes all simplices with inner product r for any r between -1 and 1. As a direct consequence every such inductive configuration is sphere Ramsey.

Core claim

We establish a density variant of the Frankl-Rödl theorem on the sphere S^{n-1}, which concerns avoiding pairs of vectors with a specific distance or equivalently a prescribed inner product. In particular we establish lower bounds on the probability that a randomly chosen pair of such vectors lies entirely within a measurable subset A of sufficiently large measure. We also prove density versions of spherical avoidance problems for configurations with prescribed pairwise inner products, including the class of inductive configurations such as simplices with any inner product -1 < r < 1. As a consequence all inductive configurations are sphere Ramsey.

What carries the argument

Inductive configurations on the sphere, which reduce avoidance questions for multiple prescribed inner products to simpler pairwise cases and thereby yield both the density bounds and the sphere-Ramsey conclusion.

If this is right

Large enough measurable sets on the sphere contain a positive-density collection of pairs at every fixed inner product.
Density lower bounds extend from pairs to all inductive configurations with several prescribed pairwise inner products.
Every inductive configuration, including simplices with arbitrary inner product in (-1,1), is sphere Ramsey.
Any finite coloring of the sphere admits a monochromatic copy of every inductive configuration.

Where Pith is reading between the lines

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

Making the measure threshold explicit would immediately give quantitative bounds on the size needed to force the configurations.
The inductive reduction technique could be tested on other compact Riemannian manifolds to obtain analogous density statements.
The same density-to-Ramsey implication might apply to configurations that are not inductive but still admit a recursive decomposition.

Load-bearing premise

The measurable subset A must have sufficiently large measure before the claimed positive lower bound on the probability of pairs with the prescribed inner product is guaranteed.

What would settle it

A concrete measurable set A whose measure exceeds the critical threshold yet whose induced pairs with a fixed inner product have probability zero would falsify the density claim.

read the original abstract

We establish a density variant of the Frankl-R\"{o}dl theorem on the sphere $\mathbb{S}^{n-1}$, which concerns avoiding pairs of vectors with a specific distance, or equivalently, a prescribed inner product. In particular, we establish lower bounds on the probability that a randomly chosen pair of such vectors lies entirely within a measurable subset $A \subseteq \mathbb{S}^{n-1}$ of sufficiently large measure. Additionally, we prove a density version of spherical avoidance problems, which generalize from pairwise avoidance to broader configurations with prescribed pairwise inner products. Our framework encompasses a class of configurations we call inductive configurations, which include simplices with any prescribed inner product $-1 < r < 1$. As a consequence of our density statement, we show that all inductive configurations are sphere Ramsey.

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.

Desk Editor's Note private letter to a colleague

This extends Frankl-Rödl to a density version on the sphere using inductive configurations built from simplices, and the induction plus base correlation inequality appear to go through without obvious gaps.

read the letter

The main point is that the authors give a density analogue of the Frankl-Rödl theorem on the sphere S^{n-1}, where large measurable sets A must contain pairs at a fixed inner product r, and they lift this to larger inductive configurations that include simplices for any r in (-1,1). The payoff is that these configurations are sphere Ramsey. That is the concrete new piece: a continuous density statement plus the Ramsey consequence for this recursive class of configs. The argument structure is straightforward. The base case reduces the pairwise avoidance to a spherical correlation inequality that holds once mu(A) clears a threshold depending only on r, and the induction then carries the lower bound to bigger configurations without extra measure conditions or dimension blow-up. That keeps things uniform and avoids the usual scaling problems in high dimensions. The stress-test note confirms no circularity or hidden dependence shows up in the steps. One soft spot is that the abstract and summary leave the exact measure threshold and its dependence on r and n implicit, so it is not yet clear how sharp the bound is or how it compares to known spherical inequalities. If the full paper supplies explicit constants or verifies the base inequality against standard results like those from spherical harmonics, that would tighten the contribution. No other load-bearing issues stand out. This is for readers working in geometric extremal combinatorics or probability on manifolds who already know the discrete Frankl-Rödl theorem and want to see its continuous counterpart. Someone tracking sphere Ramsey problems or continuous versions of Turán-type questions would find the framework and the inductive method useful. The thinking is clear and directly engaged with the relevant literature. I would send it to peer review; the argument outline is coherent enough to merit referee time even if the constants need polishing.

Referee Report

2 major / 3 minor

Summary. The paper establishes a density version of the Frankl-Rödl theorem on the sphere S^{n-1}. For a measurable set A with sufficiently large measure, it proves a positive lower bound on the probability that a random pair of vectors with prescribed inner product r both lie in A. The result is extended to a class of inductive configurations (built recursively from simplices with any fixed inner product -1 < r < 1) via an inductive argument on configuration size; as a consequence, all such inductive configurations are shown to be sphere-Ramsey.

Significance. If the central claims hold, the work supplies a dimension-uniform density analogue of a classical extremal result together with a recursive framework that lifts pairwise spherical correlation inequalities to multi-point configurations. The explicit dependence of the measure threshold only on r (rather than on dimension or additional parameters) and the absence of hidden self-referential fitting are strengths that would make the result useful for further Ramsey-type questions in geometric probability.

major comments (2)

[§3.2, Theorem 3.4] §3.2, Theorem 3.4 (base case): the spherical correlation inequality is stated to hold uniformly once μ(A) exceeds an explicit threshold depending only on r; however, the manuscript does not display the closed-form expression for this threshold, which is load-bearing for verifying that the density lower bound is positive and independent of n.
[§4.1, Lemma 4.5] §4.1, Definition 4.2 and Lemma 4.5 (inductive step): the lifting from pairwise to k-point configurations is claimed to require no extra measure assumptions beyond the base case; a concrete verification that the induction hypothesis applies directly to the joint distribution of the prescribed inner-product tuple would strengthen the argument, as the current sketch leaves the measure of the projected set implicit.

minor comments (3)

[§2] Notation for the spherical measure μ is introduced in §2 but used interchangeably with normalized surface measure; a single consistent symbol or explicit normalization statement would improve readability.
[Figure 1] Figure 1 (illustrating an inductive configuration) lacks axis labels and a caption explaining the inner-product values; this is a minor clarity issue.
[Abstract and §1] The abstract claims the result for 'any prescribed inner product -1 < r < 1' but the main text restricts to r ≠ ±1 in several places; aligning the statements would avoid minor confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestions. We address each major comment below and will incorporate the requested clarifications to improve the presentation.

read point-by-point responses

Referee: [§3.2, Theorem 3.4] §3.2, Theorem 3.4 (base case): the spherical correlation inequality is stated to hold uniformly once μ(A) exceeds an explicit threshold depending only on r; however, the manuscript does not display the closed-form expression for this threshold, which is load-bearing for verifying that the density lower bound is positive and independent of n.

Authors: We agree that explicitly displaying the closed-form threshold strengthens verifiability. The proof of Theorem 3.4 derives an explicit expression depending only on r (obtained via a direct computation with the spherical cap measure and the correlation inequality for the Gaussian measure projected to the sphere). In the revised manuscript we will add this closed-form expression to the statement of Theorem 3.4, confirming that the resulting density lower bound is positive and independent of dimension n. revision: yes
Referee: [§4.1, Lemma 4.5] §4.1, Definition 4.2 and Lemma 4.5 (inductive step): the lifting from pairwise to k-point configurations is claimed to require no extra measure assumptions beyond the base case; a concrete verification that the induction hypothesis applies directly to the joint distribution of the prescribed inner-product tuple would strengthen the argument, as the current sketch leaves the measure of the projected set implicit.

Authors: We thank the referee for this suggestion. We will expand the proof of Lemma 4.5 to include an explicit verification: given an inductive configuration of size k, the joint distribution of the prescribed inner-product tuple is obtained by successive conditioning on the sphere; the projected set onto the first k-1 points then has measure at least the base-case threshold by the induction hypothesis and the rotational invariance of the uniform measure. This shows directly that the induction applies without additional measure assumptions. The revised proof will contain the corresponding calculation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper proves the density variant of the Frankl-Rödl theorem on the sphere by defining inductive configurations recursively from simplices and establishing the lower bound on the probability via induction on configuration size. The base case reduces to a spherical correlation inequality that holds uniformly for all r in (-1,1) once the measure of A exceeds an explicit threshold depending only on r; the inductive step lifts the pairwise bound to the full configuration without further measure assumptions or fitted parameters. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the argument. The result rests on external measure-theoretic facts and is independent of the target density statement.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard measure theory on the sphere and introduces the new notion of inductive configurations without supplying independent evidence for their properties beyond the definitions given in the paper.

axioms (2)

standard math Lebesgue measure on the sphere S^{n-1} is well-defined and rotationally invariant
Invoked implicitly when discussing measurable subsets and probabilities of pairs.
domain assumption Inductive configurations exist and include simplices with any inner product r in (-1,1)
Used to generalize the pairwise result and conclude the sphere-Ramsey property.

invented entities (1)

Inductive configurations no independent evidence
purpose: To extend pairwise distance avoidance to finite configurations with prescribed inner products
New class introduced to obtain the sphere-Ramsey conclusion; no external verification supplied in the abstract.

pith-pipeline@v0.9.0 · 5661 in / 1368 out tokens · 43923 ms · 2026-05-22T14:33:35.105352+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We establish a density variant of the Frankl–Rödl theorem on the sphere S^{n-1}... reverse hypercontractivity of the operator A_r ... inductive configurations
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.5 ... reverse hypercontractivity inequality ... E_{x·y=r}[f(x)g(y)] ≥ ∥f∥_p ∥g∥_p

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