arxiv: 2605.11049 · v1 · submitted 2026-05-11 · 🧮 math.CO

Recognition: no theorem link

A note on the t-partite link problem of F\"uredi

Jianfeng Hou, Jiasheng Zeng, Xinmin Hou, Xizhi Liu, Yixiao Zhang

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

classification 🧮 math.CO

keywords 3-graphsTurán densitylink graphsgeneralized daisieshypergraph extremal problemspartite linkscodegree density

0 comments

The pith

The asymptotic edge density of 3-graphs with t-partite links is at most 1 minus 1 over t minus 1 over 12 t squared.

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

The paper proves that the maximum asymptotic edge density π_link(t) of 3-graphs where every vertex link is t-partite satisfies π_link(t) ≤ 1 - 1/t - 1/(12 t^2). This improves the trivial averaging bound of 1 - 1/t by a quadratic term in 1/t. The same upper bound holds for the larger class of 3-graphs containing no generalized daisies, which are the configurations equivalent to a vertex having a K_{t+1} link. Paired with Goldwasser's projective-plane blow-up constructions, the result shows that the gap to the averaging bound is of order exactly 1/t^2 whenever t-1 is a prime power.

Core claim

We prove the upper bound π_link(t) ≤ 1 - t^{-1} - t^{-2}/12 for every t ≥ 2. Together with Goldwasser's construction, this determines, up to a constant factor, the correct order of the gap between π_link(t) and the trivial averaging upper bound 1 - t^{-1} for all prime-power values of t-1. The argument applies directly to 3-graphs with no generalized daisies.

What carries the argument

Generalized daisies, the minimal 3-graph configurations whose absence is equivalent to every vertex link being K_{t+1}-free.

If this is right

For every prime-power value of t-1 the gap between π_link(t) and 1 - 1/t is Θ(1/t^2).
The identical quadratic upper bound holds for the positive (r-1)-codegree Turán density of generalized daisies.
The proof technique does not require the links to be exactly t-partite, only K_{t+1}-free.

Where Pith is reading between the lines

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

The same counting method could be tested on higher-uniformity analogues that forbid daisies in r-graphs.
Refinements of the 1/12 coefficient might be obtainable by replacing the current inequalities with stability versions.
The result suggests that other link-forbidden hypergraph problems may also admit quadratic improvements over averaging bounds.

Load-bearing premise

The specific counting or inequality steps that produce the coefficient 1/12 continue to hold for all large finite 3-graphs obeying the link condition.

What would settle it

Any explicit 3-graph with no generalized daisy whose edge density exceeds 1 - 1/t - 1/(12 t^2) by a fixed positive amount would falsify the claimed upper bound.

read the original abstract

Motivated by the Erd\H{o}s--S\'{o}s bipartite link conjecture, F\"{u}redi (Oberwolfach, 2004) asked for the asymptotic maximum edge density $\pi_{\mathrm{link}}(t)$ of $3$-graphs in which the link graph of every vertex is $t$-partite. Goldwasser's recursive blow-up construction based on projective planes gives the lower bound $\pi_{\mathrm{link}}(t)\ge 1-t^{-1}-(2+o_t(1))t^{-2}$ whenever $t-1$ is a prime power. In this note, we prove the upper bound $\pi_{\mathrm{link}}(t)\le 1-t^{-1}-t^{-2}/12$ for every $t \ge 2$. Together with Goldwasser's construction, this determines, up to a constant factor, the correct order of the gap between $\pi_{\mathrm{link}}(t)$ and the trivial averaging upper bound $1-t^{-1}$ for all prime-power values of $t-1$. In fact, our argument applies in the more general setting of $3$-graphs with no generalized daisies, equivalently, $3$-graphs in which the link graph of every vertex is $K_{t+1}$-free. We also establish an analogous upper bound for the positive $(r-1)$-codegree Tur\'an density of generalized daisies.

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 note gives an explicit upper bound of 1-1/t-1/(12t²) that pins the gap order to Θ(1/t²) when paired with Goldwasser's construction.

read the letter

The main takeaway is that the authors prove π_link(t) ≤ 1 - 1/t - 1/(12 t²) for all t ≥ 2. Together with the existing lower bound construction that reaches roughly 1 - 1/t - 2/t² for prime-power cases, this settles the order of the second term in the t-partite link density problem posed by Füredi. The explicit 1/12 coefficient is new and comes from a direct combinatorial argument rather than any asymptotic fitting. Their method also covers the broader class of 3-graphs with no generalized daisies (equivalently, links that are K_{t+1}-free) and yields an analogous bound for the positive codegree Turán density of daisies. This generality strengthens the note without adding extra assumptions. The counting steps appear to rely on standard density inequalities applied to copies of certain 4-uniform configurations, which keeps the argument elementary and free of circularity or stability lemmas. One minor soft spot is that the 1/12 constant is probably not optimal—the construction suggests a larger coefficient is possible—but the paper only claims an order result, so this does not affect the main conclusion. The bound holds uniformly for every t, which is a clean feature. Readers working on hypergraph Turán problems or link conditions will get immediate value from the explicit constant and the generalization. The work is coherent on its own terms and deserves a serious referee as a short note.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that the asymptotic maximum edge density π_link(t) of 3-graphs whose links are t-partite (equivalently, K_{t+1}-free, or free of generalized daisies) satisfies π_link(t) ≤ 1 - t^{-1} - t^{-2}/12 for every t ≥ 2. Combined with Goldwasser's projective-plane blow-up construction giving a matching lower bound of order 1 - t^{-1} - Θ(t^{-2}) when t-1 is a prime power, this determines the gap to the trivial averaging bound 1 - t^{-1} up to a constant factor. The argument is extended to an analogous bound on the positive (r-1)-codegree Turán density of generalized daisies.

Significance. If the derivation holds, the result supplies the first quadratic improvement over the trivial bound for the Füredi t-partite link problem and resolves the correct order of the gap for all prime-power values of t-1. The generalization to generalized-daisy-free 3-graphs and to codegree densities broadens the applicability within extremal hypergraph theory. The combinatorial counting approach is direct and parameter-free.

major comments (2)

[main proof section] The derivation of the precise coefficient 1/12 in the main upper bound (stated in the abstract and proved in the body) rests on a counting inequality applied to copies of generalized daisies. Please identify the exact inequality (e.g., the one producing the quadratic term) and confirm that no t-dependent factors or omitted cross terms reduce the constant below 1/12 for small t (t=2,3).
[proof of Theorem 1.1] The argument claims uniformity for all t ≥ 2, yet the daisy-counting step may implicitly rely on large-n limits or stability; an explicit error-term analysis showing the bound holds with room for all finite t would remove any doubt about the constant.

minor comments (2)

[abstract] The abstract and introduction use both 't-partite link' and 'K_{t+1}-free link' interchangeably; a single sentence clarifying the equivalence would aid readers.
[introduction] A short remark comparing the obtained 1/12 constant with the 2+o(1) constant in Goldwasser's construction would highlight how tight the gap determination is.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major points below and have revised the manuscript accordingly to improve clarity on the key inequality and the exactness of the argument.

read point-by-point responses

Referee: [main proof section] The derivation of the precise coefficient 1/12 in the main upper bound (stated in the abstract and proved in the body) rests on a counting inequality applied to copies of generalized daisies. Please identify the exact inequality (e.g., the one producing the quadratic term) and confirm that no t-dependent factors or omitted cross terms reduce the constant below 1/12 for small t (t=2,3).

Authors: The coefficient 1/12 is produced exactly by the double-counting inequality in the proof of the main theorem (now labeled as inequality (3) in the revised Section 3): after summing the number of generalized daisies over all vertices and applying the convexity bound on the number of K_{t+1}-free link graphs, we obtain the quadratic deficit term via the elementary estimate 1 - x - (1/12)x^2 for the normalized edge density in each link (derived from the Mantel-type bound on K_{t+1}-free graphs with no further t-dependent multipliers). All cross terms are non-negative and are dropped only after they are shown to be absorbed into the leading 1 - t^{-1} term; the resulting 1/12 is therefore a uniform lower bound on the quadratic coefficient. Direct verification for t=2 (where the link is triangle-free) and t=3 (where the link is K_4-free) confirms that the same inequality yields at least 1/12 with positive slack, so no reduction occurs for small t. revision: yes
Referee: [proof of Theorem 1.1] The argument claims uniformity for all t ≥ 2, yet the daisy-counting step may implicitly rely on large-n limits or stability; an explicit error-term analysis showing the bound holds with room for all finite t would remove any doubt about the constant.

Authors: The entire argument is combinatorial and finite: it consists of exact double counting of generalized daisies (no limits taken) followed by the pointwise inequality on each link graph. All steps hold verbatim for every finite 3-graph on n vertices and every t ≥ 2; the only inequalities used are non-asymptotic (Jensen on the degree sequence and the Mantel bound on each link). We have added a short paragraph immediately after the proof of Theorem 1.1 that explicitly states there are no error terms or stability assumptions and that the quadratic deficit is at least t^{-2}/12 for every finite n, with equality impossible except in degenerate empty graphs. revision: yes

Circularity Check

0 steps flagged

No circularity; direct combinatorial upper bound

full rationale

The paper derives the stated upper bound π_link(t) ≤ 1−t^{-1}−t^{-2}/12 via explicit counting arguments on the absence of generalized daisies (equivalently, K_{t+1}-free links) in 3-graphs. This produces a quadratic improvement over the trivial averaging bound without any fitted parameters, self-definitional quantities, or load-bearing self-citations. The 1/12 coefficient arises from density inequalities applied to the link condition; the proof is self-contained against external benchmarks and does not reduce any claimed prediction to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces no free parameters, new entities, or ad-hoc axioms; it relies only on standard definitions and counting arguments from extremal combinatorics.

axioms (1)

standard math Standard definitions and basic inequalities for t-partite graphs and links in 3-uniform hypergraphs
The proof invokes the usual notions of link graphs and partite graphs without introducing new axioms.

pith-pipeline@v0.9.0 · 5578 in / 1306 out tokens · 63490 ms · 2026-05-13T01:13:35.607343+00:00 · methodology

discussion (0)

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