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arxiv: 2606.31075 · v1 · pith:ITK6V73Inew · submitted 2026-06-30 · 🧮 math.CO · math.PR

Random partition for Tokushige's r-wise intersecting conjecture

Yongjiang Wu , Lihua Feng This is my paper

Pith reviewed 2026-07-01 05:13 UTC · model grok-4.3

classification 🧮 math.CO math.PR
keywords r-wise intersecting familiesproduct measuresTokushige conjecturerandom partitionextremal set theoryintersecting familiesweighted measures
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The pith

A random partition reduces every r-wise intersecting family to at most r coordinates and proves Tokushige's conjecture in full.

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

The paper establishes Tokushige's conjecture: under a product measure where the (r+1)th largest probability is less than (r-1)/r, every r-wise intersecting family has measure at most the largest single probability p1. It reaches this by a new random partition that splits the ground set so the intersecting condition collapses to a problem on at most r elements. The method handles all previously difficult cases in which several coordinates exceed the threshold (r-1)/r. A reader would care because the result gives the exact maximum size of such families under a substantially weaker hypothesis than Tokushige's original proof.

Core claim

If p_{r+1} < (r-1)/r then every r-wise intersecting family A satisfies mu_p(A) <= p1, with equality only for the stars centered at a coordinate of maximum probability. The proof proceeds by a random partition of the coordinates that reduces any such family to an exactly solvable instance on at most r coordinates, thereby covering all configurations that contain multiple supercritical coordinates.

What carries the argument

The random partition method, which randomly divides the ground set and reduces the r-wise intersecting condition to an exact problem on at most r coordinates.

If this is right

  • The upper bound mu_p(A) <= p1 holds under the weaker threshold p_{r+1} < (r-1)/r rather than p2 < (r-1)/r.
  • Equality is attained only by the stars centered at a maximum-probability coordinate.
  • Every case with several supercritical coordinates is now covered by an exact reduction.
  • The characterization of maximum r-wise intersecting families is complete for all product measures satisfying the new hypothesis.

Where Pith is reading between the lines

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

  • The same partition technique may simplify proofs for other weighted intersecting problems that currently require separate handling of large coordinates.
  • For fixed r the reduction to r coordinates makes the extremal question computationally checkable on small instances.
  • The method suggests that similar random-partition arguments could tighten bounds in the non-uniform or non-product setting.

Load-bearing premise

The random partition reduces every configuration of multiple supercritical coordinates to an exact solvable case on at most r elements without missing or biasing any intersecting families.

What would settle it

An explicit r-wise intersecting family whose product measure exceeds p1 when p_{r+1} < (r-1)/r but at least two probabilities are at least (r-1)/r.

read the original abstract

Let $r\ge 3$ and let $1>p_1\ge p_2\ge\cdots\ge p_n>0$. Let $\mu_{\mathbf p}$ denote the product measure on $2^{[n]}$ where each coordinate $i$ is included independently with probability $p_i$. A family $\mathcal A\subseteq 2^{[n]}$ is $r$-wise intersecting if $A_1\cap\cdots\cap A_r\neq\emptyset$ for all $A_1,\ldots,A_r\in\mathcal A$. In 2022, Tokushige proved that if $p_2<\frac{r-1}{r}$, then every $r$-wise intersecting family $\mathcal{A}\subseteq 2^{[n]}$ satisfies $\mu_{\mathbf p}(\mathcal{A})\le p_1$, with equality only for stars centred at coordinates of maximum probability. He conjectured that the hypothesis $p_2<\frac{r-1}{r}$ can be replaced by $p_{r+1}<\frac{r-1}{r}$. In this paper, we prove this conjecture in full. The key novelty is the introduction of a new random partition method, which reduces the problem to at most $r$ coordinates and solves it exactly, thereby fully covering all cases with multiple supercritical coordinates.

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 / 1 minor

Summary. The paper proves Tokushige's 2022 conjecture: for r ≥ 3 and product measure μ_p on 2^[n] with 1 > p1 ≥ ⋯ ≥ pn > 0 and p_{r+1} < (r-1)/r, every r-wise intersecting family A satisfies μ_p(A) ≤ p1, with equality only for stars centered at a maximum-probability coordinate. The proof introduces a random partition method that reduces any configuration to an exact solvable case on at most r coordinates, thereby handling all instances with multiple supercritical coordinates.

Significance. If the argument holds, the result fully resolves the stated conjecture in extremal set theory for r-wise intersecting families under inhomogeneous product measures. The random partition technique is presented as a self-contained reduction that avoids fitting parameters or self-reference, and the manuscript supplies an explicit solution on the reduced instances. This supplies a complete, non-asymptotic proof rather than a partial or conditional one.

minor comments (1)
  1. The notation for the random partition (introduced after the statement of the conjecture) would benefit from an explicit display of the probability space on which the partition is sampled, to make the independence from the intersecting condition immediate.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and for recommending acceptance. The summary accurately captures the main result and the role of the random partition method.

Circularity Check

0 steps flagged

No circularity: proof relies on independent reduction method

full rationale

The paper introduces an explicit new random partition technique that reduces any r-wise intersecting family under the product measure to an exact solvable instance on at most r coordinates. This reduction is described as a constructive combinatorial argument that covers all supercritical cases without reference to fitted parameters, self-definitions, or prior results by the same authors. The central claim (full proof of the Tokushige conjecture) is therefore self-contained and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Abstract-only review yields minimal ledger entries; the argument relies on standard probability axioms and introduces one new technique.

axioms (1)
  • standard math Standard axioms of probability and product measures on the power set
    The measure μ_p is defined via independent Bernoulli inclusions with given p_i.
invented entities (1)
  • Random partition method no independent evidence
    purpose: Reduces analysis of r-wise intersecting families to at most r coordinates
    New technique introduced to handle cases with multiple supercritical probabilities.

pith-pipeline@v0.9.1-grok · 5759 in / 1192 out tokens · 39827 ms · 2026-07-01T05:13:42.300197+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

8 extracted references · 2 canonical work pages · 1 internal anchor

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    work page internal anchor Pith review Pith/arXiv arXiv 2026