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arxiv: 2606.26521 · v1 · pith:LWDCAMFPnew · submitted 2026-06-25 · 🧮 math.CO

The Suda-Tanaka-Tokushige conjecture for p-biased intersecting families

Yongjiang Wu , Lihua Feng This is my paper

Pith reviewed 2026-06-26 04:44 UTC · model grok-4.3

classification 🧮 math.CO
keywords t-intersecting familiesproduct measuresbiased measuresextremal set theorySuda-Tanaka-Tokushige conjecture
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The pith

Under p_{t+2} ≤ 1/(t+1) every t-intersecting family has μ_p measure at most the product of the first t biases.

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

The paper proves the Suda-Tanaka-Tokushige conjecture in a strengthened t-intersecting version. For ordered biases satisfying the threshold p_{t+2} ≤ 1/(t+1), the product measure of any t-intersecting subfamily of the power set is at most the product of the t largest bias values. The bound is achieved exactly by the families consisting of all sets that contain a fixed t-element set whose bias product equals that maximum. A reader would care because the result determines the precise maximum measure attainable by t-intersecting families and recovers several classical extremal theorems as special cases.

Core claim

For any t ≥ 1, if 1 > p_1 ≥ ⋯ ≥ p_n > 0 and p_{t+2} ≤ 1/(t+1), then every t-intersecting family A ⊆ 2^[n] satisfies μ_p(A) ≤ ∏_{i=1}^t p_i. Moreover, when p_{t+2} < 1/(t+1), equality holds if and only if A equals the collection of all sets containing some fixed T ∈ inom{[n]}{t} whose bias product equals the maximum possible such product.

What carries the argument

the non-uniform product measure μ_p defined by μ_p(A) = ∑_{B∈A} (∏_{i∈B} p_i) (∏_{j∉B} (1-p_j)), restricted to t-intersecting subfamilies

If this is right

  • The original 2017 conjecture for ordinary intersecting families (t=1) holds under the stated bias ordering and p_3 ≤ 1/2.
  • When the inequality on p_{t+2} is strict, the only families attaining the bound are the t-stars centered at a t-set of maximum bias product.
  • The result recovers the classical theorems of Fishburn-Frankl-Freed-Lagarias-Odlyzko and of Friedgut as special cases.
  • The bound is independent of n and applies uniformly once the threshold condition on the biases is met.

Where Pith is reading between the lines

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

  • The same threshold technique may extend to uniform t-intersecting families of fixed size k.
  • When p_{t+2} exceeds 1/(t+1) the extremal examples may switch to other constructions such as taking all sets meeting a larger fixed set.
  • Small-n computational enumeration could verify whether the threshold 1/(t+1) is sharp by exhibiting families that exceed the product bound precisely when the inequality is reversed.

Load-bearing premise

The biases satisfy the decreasing ordering 1 > p_1 ≥ ⋯ ≥ p_n > 0 together with the numerical threshold p_{t+2} ≤ 1/(t+1).

What would settle it

A concrete t-intersecting family whose μ_p measure strictly exceeds ∏_{i=1}^t p_i while p_{t+2} ≤ 1/(t+1) still holds would falsify the bound.

read the original abstract

In 2017, Suda, Tanaka and Tokushige conjectured that if $1>p_1\ge\cdots\ge p_n>0$ with $p_3\le \frac{1}{2}$, then every intersecting family $\mathcal A\subseteq 2^{[n]}$ satisfies $\mu_{\mathbf{p}}(\mathcal A)\le p_1$, where $\mu_{\mathbf{p}}$ is the non-uniform product measure defined by $\mu_{\mathbf{p}}(\mathcal{A})=\sum_{A\in\mathcal{A}} \prod_{i\in A} p_i \prod_{j\in [n]\setminus A}(1-p_j)$. In addition, if $p_1 > p_3$ or $p_1 < \frac{1}{2}$, then equality holds if and only if $\mathcal{A}$ is a star centered at some $i \in [n]$ with $p_i = p_1$. In this paper, we prove this conjecture in the following stronger $t$-intersecting form: for any $t\ge 1$, if $p_{t+2}\le \frac{1}{t+1}$, then every $t$-intersecting family $\mathcal{A} \subseteq 2^{[n]}$ satisfies $\mu_{\mathbf{p}}(\mathcal A)\le \prod_{i=1}^t p_i$. Moreover, when $p_{t+2}<\frac{1}{t+1}$, equality holds if and only if $\mathcal{A}=\{A\subseteq [n]: T\subseteq A\}$ for some $T\in \binom{[n]}{t}$ with $\prod_{i\in T} p_i=\prod_{i=1}^t p_i$. Our result unifies and generalizes the classical theorems of Fishburn-Frankl-Freed-Lagarias-Odlyzko and Friedgut.

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 / 2 minor

Summary. The manuscript proves the Suda-Tanaka-Tokushige conjecture in the following stronger form: for any t ≥ 1, if the biases satisfy 1 > p1 ≥ ⋯ ≥ pn > 0 and p_{t+2} ≤ 1/(t+1), then every t-intersecting family A ⊆ 2^[n] satisfies μ_p(A) ≤ ∏_{i=1}^t p_i, with the equality case characterized when the threshold is strict as the t-star centered at a t-set T achieving the maximal product.

Significance. The result directly proves the 2017 conjecture while unifying and generalizing the classical theorems of Fishburn-Frankl-Freed-Lagarias-Odlyzko and Friedgut. The argument combines shifting/compression with an inductive reduction to the t=1 case (already known), and explicitly handles the equality characterization under the given threshold; these are load-bearing strengths of the manuscript.

minor comments (2)
  1. [Abstract] The abstract states the equality condition as '∏_{i∈T} p_i = ∏_{i=1}^t p_i' but does not explicitly note that this selects the t-set of largest product among those with the given biases; a parenthetical clarification would aid readability.
  2. [Introduction] The introduction could include a one-sentence pointer to the precise location of the t=1 base case (Fishburn-Frankl et al. or the original conjecture) to make the inductive step easier to trace.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. We are grateful for the recommendation to accept and for noting that the result proves the 2017 conjecture while unifying and generalizing the theorems of Fishburn-Frankl-Freed-Lagarias-Odlyzko and Friedgut.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The manuscript proves the external Suda-Tanaka-Tokushige conjecture (and its t-intersecting generalization) via shifting/compression plus induction on the t=1 case, which is cited as already known from the 2017 conjecture or Fishburn-Frankl et al. The ordering 1 > p1 ≥ ⋯ ≥ pn > 0 and the threshold p_{t+2} ≤ 1/(t+1) are stated hypotheses, not internally derived or fitted. No equation reduces by construction to a fitted parameter, no load-bearing uniqueness theorem is imported from the present authors, and no ansatz is smuggled via self-citation. The equality characterization follows directly from the same compression step under the strict inequality. The central claim therefore rests on independent combinatorial arguments rather than self-referential definitions or renamings.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure mathematical proof relying on standard combinatorial techniques and the definition of the product measure; no free parameters or invented entities appear in the abstract statement.

axioms (1)
  • domain assumption Basic properties of product measures on the power set and the definition of t-intersecting families
    The theorem statement invokes the product measure μ_p and the t-intersecting property as given.

pith-pipeline@v0.9.1-grok · 5881 in / 1458 out tokens · 59705 ms · 2026-06-26T04:44:02.381902+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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

  1. Random partition for Tokushige's $r$-wise intersecting conjecture

    math.CO 2026-06 unverdicted novelty 7.0

    Proves Tokushige's conjecture on r-wise intersecting families under product measures by replacing p2 < (r-1)/r with the weaker p_{r+1} < (r-1)/r via a random partition method.

Reference graph

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