Extremal set theory · combinatorics · transversal codes

Erdős–Rado Sunflower Conjecture

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For three petals, must every sufficiently large uniform set family contain three sets with the same pairwise intersection?

\exists C < \infty \forall r \geq 1, f_{3} (r) \leq C^{r}

Erdős Problem #20

Known results and sources

Three petals share one core: the geometric pattern whose unavoidable appearance is still conjectural at exponential scale.

Research problem

Exact mathematical statement

Let $f_3(r)$ be the largest size of an $r$ -uniform family with no three-petal sunflower. The conjecture is

\exists C<\infty\;\forall r\ge 1,\qquad f_3(r)\le C^r.

Only the three-petal form is in scope. The handoff does not claim a reduction to every fixed number of petals.

Problem infographic

Problem at a glance

A deep forest-green scientific plate explains the three-petal Erdős–Rado sunflower conjecture. One antique-gold core R touches three nonoverlapping petals P₁, P₂, and P₃, with A=R∪P₁, B=R∪P₂, C=R∪P₃ and A∩B=A∩C=B∩C=R. A secondary vignette labeled BINARY TRANSVERSAL CODE shows binary words of length r, {0,1}ʳ, 2ʳ codewords, and the statement sunflower-free in the code model. The exact question asks whether some finite C satisfies f₃(r)≤Cʳ for every r, and the plate states that the general case remains open. — Three r-element sets form a three-petal sunflower when their three pairwise intersections are the same core R, equivalently when their petals outside R are pairwise disjoint. The right vignette is a binary transversal-code baseline: {0,1}ʳ has 2ʳ codewords and is sunflower-free in the code model; interpreting each word x as its transversal r-set Aₓ gives the corresponding r-uniform construction. The conjecture asks whether every sunflower-free r-uniform family nevertheless has size at most Cʳ for one absolute finite C. The general case remains open.

Current mathematical picture

Where work on Erdős–Rado Sunflower Conjecture stands

Recent proof claim under review

The current research map records the three-petal statement, the transversal and pairwise-agreeing reductions, representative structural closure theorems, the live binomial-trace recurrence and exact geometric telescoping, adversarial models that refute several shortcuts, and the boundary-stability/direct-sum frontier. The current work reports project proofs and verified computations but no independent peer review or accepted proof; the conjecture remains unresolved.

Strongest supported footholdStructural regimes bounded

The current work proves exponential or polynomial bounds in dense-box, laminar, bounded-width, bounded-agreement, and combined width-agreement regimes.

Evidence posture · Reported result

Leading routeIsosceles traces and binomial moments

This is the source's preferred route: use strict root-label growth to obtain BTR, align scales so interior moments telescope, and close the remaining boundary terms with code structure.

Route status · Active route

Useful failureSharp collision tensorization

The one-coordinate collision inequality, sharp spectral kernel bound, and sharp uniform energy bound are refuted by the retained correlated 20-word example and type-class amplification.

Route status · Refuted route

Main reductionBinomial-trace recurrence derived

Weighted lower and recursive upper bounds for rooted isosceles triples combine into the current work's strongest current recurrence.

Evidence posture · Reported reduction

Completed special caseBounded maximum-agreement theorem

If every pair in a pairwise-agreeing sunflower-free code agrees in at most t coordinates, then its size is at most Σⱼ≤min(t,r) binom(r,j)aⱼ and hence at most 2(2r)ᵗ for t≥1.

Evidence posture · Source-reported route statement

Priority open bridgeProve boundary stability for BTR

Control the explicit right-boundary moments left by the aligned telescoping identity strongly enough to contradict the summed BTR inequality whenever M>Aʳ.

Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Erdős–Rado Sunflower Conjecture in numbers

813retained lines of mathematical investigation813 in the current working snapshot

Argument development: 697 · 86%
Explored or eliminated routes: 24 · 3%
Computational analysis: 17 · 2%
Open obligations: 23 · 3%
Definitions and setup: 52 · 6%

17selected mapped statements9routes investigated7reported milestones6open questions6contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byPaul Erdős, Richard RadoProposed1960Status checkedAug 2, 2026

Source recordedAug 2, 2026

Recommended next task

Prove boundary stability for BTR

Control the explicit right-boundary moments left by the aligned telescoping identity strongly enough to contradict the summed BTR inequality whenever M>Aʳ.

Suggested move: Normalize the agreement-size distribution, solve moderate-rank BTR feasibility problems on an aligned geometric scale grid, and inspect dual certificates for an asymptotically stable pattern.

What would count as progress

Give a source-auditable inequality that bounds all surviving boundary terms without increasing the induction base.
Show that its combination with BTR contradicts M>Aʳ under lower-rank induction.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

24 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Active routeIsosceles traces and binomial moments

This is the source's preferred route: use strict root-label growth to obtain BTR, align scales so interior moments telescope, and close the remaining boundary terms with code structure.

Route status · Active route

Active routeCross-class direct-sum inequality

A structural gain across exact root classes is active as the equivalent combinatorial formulation of boundary stability.

Route status · Active route

Explored alternatives

Other routes

7 recorded

Narrowed routeCharacteristic-three dense-box route

The route is valid within a dense minimal coordinate box, but the prefix construction refutes the universal projection lemma intended to force arbitrary codes into that regime.

Route status · Narrowed route

Narrowed routeBinary linear testbed

The exact rank-one criterion remains a live structured subproblem, with O(r log r) known in the current work and m≤Cr as the open target; a nonlinear transfer would still be required.

Route status · Narrowed route

Refuted routeSharp collision tensorization

The one-coordinate collision inequality, sharp spectral kernel bound, and sharp uniform energy bound are refuted by the retained correlated 20-word example and type-class amplification.

Route status · Refuted route

Browse 4 more explored routes

Refuted routeUniversal heavy-fiber induction

Random high-rank linear examples have arbitrarily small coordinate fibers, so a universal constant-density fiber cannot drive induction.

Route status · Refuted route

Useful but insufficientIndependent trace-class induction

Separate bounds lose the cross-class coupling, reproduce factorial branching, and fail even under exact two-pivot bookkeeping at fixed induction base.

Route status · Useful but insufficient

Not yet justifiedBinary coarsening

The two-to-one version is not justified because its asserted counterexample lacks retained bytes and a certificate; the power-preserving one-bit version remains open but parked and is stronger than the conjecture.

Route status · Not yet justified

Route held in reserveHigh-rate trace entropy

A size-sensitive high-rate trace-entropy lemma is a valid conditional closure, but it contains the same small-core direct-sum obstruction as the preferred BTR route.

Route status · Route held in reserve

Route statements and reductions

Statements the next route can inspect and build on

Route statementStrict root-label growth

If two words y and z have the same exact equality label S relative to a root x, then S is a strict subset of σ(y,z).

Source-reported route statement

Route statementBinomial-trace recurrence

Under I(d) ≤ Aᵈ for every d<r, the normalized pair-agreement moment P obeys (M−1)/Zᵣ(u)−P(u) ≤ Aʳ[P(1+u/A)−1−P(u/A)] for every u>0.

Source-reported route statement

Route statementAligned geometric telescoping

Choosing A=(1+1/A)ᵏ and geometric scales uⱼ=u₀(1+1/A)ʲ cancels every interior weighted pair moment in the summed recurrence and leaves only explicit boundary terms.

Source-reported route statement

Route statementCross-class direct-sum gain

A viable closing theorem must gain across many exact root classes while surviving the binary cube, prefix, Kₘ-star, many-singleton-branch, and random linear adversarial models.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

6 featured tasks

Prove boundary stability for BTR

Control the explicit right-boundary moments left by the aligned telescoping identity strongly enough to contradict the summed BTR inequality whenever M>Aʳ.

Ready to work on

Prove a cross-class direct-sum gain

Find a gain across exact root classes that charges total residual information and survives all retained hierarchical, many-branch, and high-rank adversarial models.

Suggested move: Test each proposed inequality against the binary cube, prefix code, Kₘ-star, many-singleton-branch, and random linear families before attempting induction.

Ready to work on

Extract an asymptotic BTR dual certificate

Find a positive combination of finite BTR, lower-tail, factorial-moment, and exact combinatorial constraints whose coefficients telescope or have bounded total mass uniformly in rank.

Suggested move: Run finite feasibility experiments for k=2 and nearby scale windows, then symbolically identify a dual pattern stable in r.

Ready to work on

Strengthen endpoint inequalities

Use pairwise agreement, strict root-class growth, equivalence-relation structure, and conditional width/agreement bounds to control terminal P(v) by lower-scale moments.

Suggested move: Derive a boundary-moment inequality that couples multiple exact root classes instead of applying the standalone factorial envelope.

Ready to work on

Audit retained theorems and finite examples

Before publication-grade use, independently check the project-proved claims in Sections 4–5, rerun the retained exact finite-example script, and recover or replace the missing 18-word certificate before revisiting binary coarsening.

Suggested move: Perform an independent theorem audit and run Appendix B with exact rational arithmetic; do not cite the missing 18-word computation without reconstruction.

Ready to work on

Close the linear testbed at m≤Cr

Improve the retained O(r log r) dimension bound under the exact two-plane rank-one witness criterion to a linear bound m≤Cr.

Suggested move: Study projective-line covers, exterior-square formulations, kernel-lattice submodularity, or a nested-versus-transverse map decomposition.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 2, 2026

Current statusRecent proof claim under review

The maintained Erdős Problems entry and a 2026 peer-reviewed survey treat the conjecture as open. A June 2026 arXiv preprint claims a proof; no independent validation or peer-reviewed acceptance was located in this review, so the status is not promoted.

[10][5][3]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2021
Authoritative summaryRefinements by Rao and by Bell-Chueluecha-Warnke yield the current surveyed record f(n,k) < (C k log n)^n for an absolute C > 1.[10][5]
2019
PreprintAlweiss, Lovett, Wu, and Zhang introduced the robust-sunflower breakthrough, giving a bound of order (w^3 log n log log n)^n in the survey's notation.[5][1]
1997
Peer reviewedKostochka improved the factorial bound for three petals by a subfactorial factor involving log log log n / log log n.[5]
1960
Peer reviewedErdős and Rado proved the factorial sunflower bound f(n,k) <= (k-1)^n n! up to the conventional threshold offset.[6][8]

Mathematical neighborhood

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

7 mapped milestonesretained argument map

Browse all 7 mapped stages

stage 1Transversal and pairwise-agreeing reduction
stage 2Structural regimes and linear testbed
stage 3Binomial-trace recurrence
stage 4Boundary-stability frontier isolated
stage 5Collision tensorization refuted
stage 6Adversarial models narrow the route
stage 7Finite-duality and direct-sum agenda

Transversal and pairwise-agreeing reductionThe current work reduces the three-petal set-family conjecture to exponential control of pairwise-agreeing sunflower-free transversal codes.