Erdős Problem #20

Problem 20 (Sunflowers): Round 3 --- extremal structure closure for the $n=2$ slice

1) ROUND-3 OBJECTIVE

Path (A) inside the nontrivial special slice already opened in Round 2.

Round 2 determined the exact value of $f(2,k)$ for every $k\ge 2$, but it stopped at the level of the numerical extremal value. The remaining structural gap in that slice is to determine all extremal $2$-uniform families of size $f(2,k)-1$.

In this round I close that gap completely: I prove a full extremal-structure theorem for the entire $n=2$ slice. In particular:

• for odd $k$, the extremal family is unique up to relabeling;
• for even $k>2$, I characterize all extremal families by their degree sequence on the support, and I exhibit two explicit nonisomorphic extremal constructions.

This is the most promising continuation because it strictly strengthens Round 2's exact formula into a complete classification theorem for the same infinite family of cases.

2) Round-2 FOUNDATION USED

I use only the following Round 2 results.

• Round 2 Lemma 20.4: for $2$-sets, a $k$-sunflower is equivalent to either a matching of size $k$ or a star $K_{1,k}$.
• Round 2 Theorem 20.5: for every $k\ge 2$, \[ f(2,k)= \begin{cases} k(k-1)+1, & k\ \text{odd},\\[3pt] \left\lfloor \dfrac{(k-1)(2k-1)}{2}\right\rfloor+1, & k\ \text{even}. \end{cases} \] Equivalently, if \[ M(k):=\max\{|E(G)|:\Delta(G)\le k-1,\ \nu(G)\le k-1\}, \] then $f(2,k)=M(k)+1$.
• Round 2 explicit constructions: the odd case $K_k\cup K_k$, and the even-case circulant construction on $2k-1$ vertices.

I do not reprove these facts.

3) NEW INSIGHT / TOOL (ROUND-3)

New tool: the equality characterization in the Abbott--Hanson--Sauer theorem.

Round 2 used only the numerical upper bound from Abbott--Hanson--Sauer. The new ingredient here is the extremal-case classification (as stated in Theorem 3.2 of [KupavskiiNoskov2025]): once equality holds, the extremal graph is either

• exactly $K_k\cup K_k$ when $k$ is odd, or
• when $k$ is even and $k>2$, a graph on $2k-1$ vertices with one vertex of degree $k-2$ and all other vertices of degree $k-1$.

A second new point is the support reduction: for a $2$-uniform family $\mathcal F$, viewing its support graph on $\bigcup\mathcal F$ automatically removes isolated vertices, which is exactly the hypothesis needed to apply the equality case.

4) ATTACK PLAN (ROUND-3)

Gap left by Round 2. Round 2 computed the exact number $f(2,k)$, but it did not determine whether the extremal family is unique, nor did it characterize all equality cases.

Claims proved in this round.

1. Every extremal $k$-sunflower-free family of $2$-sets is completely classified.
2. For odd $k$, the extremal family is unique up to isomorphism.
3. For even $k>2$, extremal families are not unique; indeed there are at least two explicit nonisomorphic extremal families for every even $k\ge 4$.

Why this overcomes the previous obstacle. The only missing ingredient after Round 2 was structural, not numerical. Equality in Theorem 3.2 of [KupavskiiNoskov2025] supplies exactly the missing structure.

5) WORK (ROUND-3)

Setup.

Let $\mathcal F$ be a family of $2$-element sets, and let $G$ be its support graph: \[ V(G)=\bigcup\mathcal F, \qquad E(G)=\mathcal F. \] Then $G$ has no isolated vertices by definition.

By Round 2 Lemma 20.4, $\mathcal F$ is $k$-sunflower-free if and only if \[ \Delta(G)\le k-1 \qquad\text{and}\qquad \nu(G)\le k-1. \] If moreover $|\mathcal F|=f(2,k)-1$, then Round 2 Theorem 20.5 gives \[ |E(G)|=M(k). \] So extremal $2$-uniform sunflower-free families are exactly the edge sets of graphs achieving equality in the graph extremal problem defining $M(k)$.

Theorem 20.7 (Abbott--Hanson--Sauer with equality structure).

Let $s\ge 2$. Let $G$ be a graph with no isolated vertices, no matching of size $s$, and no vertex of degree $s$. Then \[ |E(G)|\le \begin{cases} s(s-1), & s\ \text{odd},\\[3pt] (s-1)^2+\dfrac{s-2}{2}, & s\ \text{even}. \end{cases} \] Moreover, if equality holds, then:

1. if $s$ is odd, $G\cong K_s\cup K_s$;
2. if $s$ is even and $s>2$, then $|V(G)|=2s-1$ and the degree multiset is \[ \{s-2,(s-1)^{(2s-2)}\}; \]
3. if $s=2$, then $G\cong K_2$.

This is exactly Theorem 3.2 of [KupavskiiNoskov2025], where the equality characterization is stated under the standing assumption that $G$ has no isolated vertices.

Theorem 20.8 (Full extremal classification for $f(2,k)$).

Let $k\ge 2$. Let $\mathcal F$ be a family of $2$-sets with no $k$-sunflower and with \[ |\mathcal F|=f(2,k)-1. \] Let $G$ be its support graph. Then exactly one of the following holds.

1. If $k=2$, then $G\cong K_2$.
2. If $k$ is odd, then \[ G\cong K_k\cup K_k. \] In particular $|V(G)|=2k$, and the extremal family is unique up to relabeling.
3. If $k$ is even and $k>2$, then \[ |V(G)|=2k-1, \qquad \deg_G(v)=k-1\ \text{for }2k-2\text{ vertices}, \qquad \deg_G(u)=k-2\ \text{for one vertex }u. \] Conversely, every graph with this degree multiset is $k$-sunflower-free and has exactly $f(2,k)-1$ edges.

Proof. Because $\mathcal F$ is $k$-sunflower-free, Round 2 Lemma 20.4 gives \[ \Delta(G)\le k-1, \qquad \nu(G)\le k-1. \] Because $|\mathcal F|=f(2,k)-1=M(k)$, the graph $G$ attains equality in the extremal problem for $M(k)$. Since $G$ has no isolated vertices (it is the support graph), Theorem 20.7 applies with $s=k$.

If $k=2$, Theorem 20.7 gives $G\cong K_2$.

If $k$ is odd, Theorem 20.7 gives $G\cong K_k\cup K_k$.

If $k$ is even and $k>2$, Theorem 20.7 gives that $G$ has $2k-1$ vertices and degree multiset \[ \{k-2,(k-1)^{(2k-2)}\}. \] This proves the forward implication.

For the converse in the even case, let $G$ be any graph on $2k-1$ vertices with one vertex of degree $k-2$ and all other vertices of degree $k-1$. Then $\Delta(G)=k-1$, and also $\nu(G)\le k-1$ because $|V(G)|=2k-1<2k$. Therefore, by Round 2 Lemma 20.4, $E(G)$ is $k$-sunflower-free.

Finally, by the handshake lemma, \[ |E(G)|=\frac{(k-2)+(2k-2)(k-1)}{2} =(k-1)^2+\frac{k-2}{2} =\left\lfloor \frac{(k-1)(2k-1)}{2}\right\rfloor =f(2,k)-1, \] where the last equality is Round 2 Theorem 20.5. Hence every such graph is extremal. $\square$

Corollary 20.9 (Parity dichotomy on uniqueness).

For the $n=2$ slice, the extremal family of size $f(2,k)-1$ is unique up to isomorphism when $k$ is odd, and nonunique when $k$ is even and greater than $2$.

Proof. The odd case is immediate from Theorem 20.8. To show nonuniqueness for even $k>2$, it suffices to give two nonisomorphic extremal constructions. This is done in Proposition 20.10 below. $\square$

Proposition 20.10 (Two explicit nonisomorphic extremal families for each even $k\ge 4$).

Fix an even integer $k\ge 4$. Then there exist at least two nonisomorphic extremal $k$-sunflower-free families of $2$-sets.

Proof. We describe two support graphs.

Construction A (circulant type). Take the even-case graph from Round 2: on $\mathbb Z/(2k-1)\mathbb Z$, let $H$ be the circulant graph with step set \[ \{\pm 1,\pm 2,\dots,\pm (k/2-1)\}, \] and let $C$ be the cycle consisting of the step-$k/2$ edges. Add to $H$ a maximum matching $M$ of $C$ leaving one vertex, say $0$, uncovered. Call the resulting graph $G^{\mathrm{circ}}_k$.

Round 2 already proved that $G^{\mathrm{circ}}_k$ is extremal.

Construction B (bipartite type). Let $A=\{a_1,\dots,a_{k-1}\}$ and $B=\{b_1,\dots,b_{k-1}\}$, and start with the complete bipartite graph $K_{k-1,k-1}$ on $A\sqcup B$. Let \[ M_0=\{a_i b_i:1\le i\le (k-2)/2\} \] be a matching of size $(k-2)/2$. Delete the edges of $M_0$. Then add one new vertex $u$ and join $u$ to all $2\cdot (k-2)/2=k-2$ endpoints of the deleted matching. Call the resulting graph $G^{\mathrm{bip}}_k$.

We now verify that $G^{\mathrm{bip}}_k$ is extremal. The new vertex $u$ has degree $k-2$. Every endpoint of a deleted matching edge loses one edge in $K_{k-1,k-1}$ and gains the edge to $u$, so its degree is again $k-1$. Every other old vertex still has degree $k-1$. Thus $G^{\mathrm{bip}}_k$ has degree multiset \[ \{k-2,(k-1)^{(2k-2)}\}. \] By Theorem 20.8, it is extremal.

It remains to prove that the two extremal graphs are not isomorphic. In each graph there is a unique vertex of degree $k-2$; call it $u_{\mathrm{circ}}$ in $G^{\mathrm{circ}}_k$ and $u_{\mathrm{bip}}$ in $G^{\mathrm{bip}}_k$.

Now, \[ G^{\mathrm{bip}}_k-u_{\mathrm{bip}}=K_{k-1,k-1}-M_0, \] which is bipartite.

On the other hand, $G^{\mathrm{circ}}_k-u_{\mathrm{circ}}$ is not bipartite. Indeed:

• if $k\ge 6$, then the vertices $1,2,3$ form a triangle, because the differences $1$ and $2$ belong to the step set of $H$;
• if $k=4$, then $G^{\mathrm{circ}}_4-u_{\mathrm{circ}}$ contains the triangle $\{2,3,4\}$: the edges $23$ and $34$ lie on the base $7$-cycle, while $24$ is one of the added matching edges.

So one deletion graph is bipartite and the other is not; hence $G^{\mathrm{circ}}_k\not\cong G^{\mathrm{bip}}_k$.

Therefore there are at least two nonisomorphic extremal families for every even $k\ge 4$. $\square$

Consequence for the global problem.

Round 2 already ruled out any counterexample strategy living purely in $2$-uniform families. Round 3 strengthens this: the full $2$-uniform extremal picture is now completely understood, including the parity-dependent uniqueness/nonuniqueness dichotomy. In particular, any future attack on the full sunflower conjecture cannot hope to extract a single parity-blind "canonical" extremal $2$-uniform core.

6) ADVERSARIAL VERIFICATION

(i) Isolated-vertex hypothesis in Theorem 20.7. The equality characterization quoted from [KupavskiiNoskov2025] assumes no isolated vertices. This is harmless here because I work with the support graph $V(G)=\bigcup\mathcal F$, which has no isolated vertices by definition.

(ii) Converse in the even case. I checked that no hidden matching argument is needed: any graph on $2k-1$ vertices is automatically free of a matching of size $k$. Thus degree sequence alone really suffices.

(iii) Boundary cases. For $k=2$, Theorem 20.8 returns the unique one-edge family. For $k=3$, it recovers the Round 1/2 extremal graph $K_3\cup K_3$. For $k=4$, Proposition 20.10 gives two concrete $10$-edge extremal graphs on $7$ vertices.

(iv) Nonisomorphism certificate in Proposition 20.10. The certificate uses the unique degree-$(k-2)$ vertex in each graph, so it is isomorphism-invariant. Deleting that vertex produces a bipartite graph in Construction B and a nonbipartite graph in Construction A.

(v) Reference correction. Round 2 cited arXiv:2511.17142 as a single-author paper "Erdős's forbidden sunflower problem." The checked source is the two-author paper Exact results and the structure of extremal families for the Duke--Erdős forbidden sunflower problem by Kupavskii and Noskov; this is the source used here for Theorem 3.2.

7) FINAL (EXACTLY ONE)

UNRESOLVED (BUT STRICTLY ADVANCED).

The original Erdős--Rado sunflower conjecture remains open. However, the $n=2$ slice is now closed at the structural level as well as the numerical level: not only is $f(2,k)$ known exactly for every $k\ge 2$, but all extremal $2$-uniform families of size $f(2,k)-1$ are now fully classified.

8) COMPLETION ESTIMATE

COMPLETION: 38%

9) REFERENCES

References

• [AHS72] H. L. Abbott, D. Hanson, and N. Sauer, Intersection theorems for systems of sets, Journal of Combinatorial Theory, Series A 12 $1972$, 381--389.
• [KupavskiiNoskov2025] A. Kupavskii and F. Noskov, Exact results and the structure of extremal families for the Duke--Erdős forbidden sunflower problem, arXiv:2511.17142 $2025$.
• [Rao2025] A. Rao, The Story of Sunflowers, preprint, September 2025.

OBJECTIVE AND SOURCE REVIEW

Erdős #20; independent attempt, 2026-09-23. For fixed $k\ge3$, the sunflower conjecture asks for $f(n,k)\le C_k^n$, where $f(n,k)$ is the least forcing size for a $k$-sunflower among distinct $n$-sets. I read all of the supplied 20.tex, which classifies the $n=2$ extremizers using an external theorem. I checked Theorem 3.2 of Kupavskii--Noskov. Here I investigate how much that fixed-rank information propagates when the rank grows.

WORK: EXACT PRODUCT OBSTRUCTION

Let $\mathcal A$ and $\mathcal B$ be uniform $k$-sunflower-free families on disjoint ground sets. Their block product \[ \mathcal A\otimes\mathcal B=\{A\cup B:A\in\mathcal A,\ B\in\mathcal B\} \] is $k$-sunflower-free. Indeed, suppose $A_i\cup B_i$, $1\le i\le k$, have all pairwise intersections equal. Projecting onto the first block gives equal pairwise intersections of the $A_i$. If two $A_i$ coincide, their intersection equals this entire set; every other $A_j$ contains it and, by uniformity, equals it. Otherwise all $A_i$ are distinct and form a forbidden sunflower. Thus every $A_i$ is equal. The identical argument makes every $B_i$ equal, contradicting distinctness of the original $k$ product sets.

This proves, writing $M(n,k)=f(n,k)-1$, \[ M(n+m,k)\ge M(n,k)M(m,k). \] Combining the supplied two-element extremizers with $k-1$ distinct singletons yields \[ M(n,k)\ge M(2,k)^{\lfloor n/2\rfloor}(k-1)^{n\bmod2}, \] where $M(2,k)=k(k-1)$ for odd $k$ and $(k-1)^2+(k-2)/2$ for even $k$. In particular any proposed exponential upper-bound constant must satisfy $C_k\ge\sqrt{M(2,k)}$. This block-product observation is standard and is not claimed as a new lower bound.

ADVERSARIAL VERIFICATION AND REMAINING GAP

The projection argument must allow repeated projected sets; merely saying that a sunflower projects to a sunflower would be incorrect. Uniformity closes precisely that loophole. Disjoint blocks make the product map injective. The argument provides a lower bound, whereas the conjecture needs an upper bound. The complete $n=2$ classification does not imply that high-rank extremizers split into blocks. Establishing such a decomposition, or a substitute controlling families with strongly coupled coordinates, remains wholly open in this attempt.

SOURCES CHECKED

Complete local source: gpt_pro_5.4/20.tex. Primary source checked: Kupavskii--Noskov, Theorem 3.2, https://arxiv.org/html/2511.17142v1. The checked theorem is used only for the already known value of $M(2,k)$; the product lemma is proved above.

FINAL

UNRESOLVED. The fixed-rank results give a necessary exponential base, not the conjectured upper bound. No new general upper bound was obtained.

COMPLETION ESTIMATE

Conservative subjective progress toward the full rank-uniform conjecture. COMPLETION: 8%

Problem 20 (Sunflowers): fixed-window entropy ledger framework

1) FORMAL RESTATEMENT

For integers $n\ge 1$ and $k\ge 2$, a $k$-sunflower is a collection of distinct sets $S_1,\ldots,S_k$ for which there is a common kernel $K$ such that \[ S_i\cap S_j=K\qquad (i\ne j). \] Let $f(n,k)$ be the least integer such that every $n$-uniform family $\mathcal F$ with $|\mathcal F|\ge f(n,k)$ contains a $k$-sunflower. Problem 20 asks whether, for every fixed $k$, there is a constant $C_k$ such that \[ f(n,k)\le C_k^n\qquad\text{for all }n. \]

This attempt focuses on the three-petal case after the standard reduction to balanced two-collision codes. It does not claim a complete proof of the sunflower conjecture.

2) QUICK LITERATURE/CONTEXT CHECK

The classical Erdos--Rado sunflower lemma gives a superexponential-in-$n$ bound for fixed $k$. The modern line of work of Alweiss--Lovett--Wu--Zhang, Rao, Tao, and Bell--Chueluecha--Warnke gives the current logarithmic-base improvements through spread or robust-sunflower methods. Entropy arguments are therefore already standard in this problem. Collision estimates are also standard local tools, and potential/amortized accounting is a standard proof technique.

The proposed novelty of this attempt is architectural rather than a new standalone inequality. It asks whether the reusable spread invariant in ALWZ/Rao/Tao-type arguments can be replaced by a coordinate-level collision/accounting invariant.

3) ATTACK PLAN

Reduction target. Reduce the three-petal sunflower problem to bounding balanced two-collision codes: \[ \mathcal C\subseteq \prod_{i=1}^n \Omega_i, \] where every three distinct codewords have a coordinate in which exactly two of the three symbols agree. A bound $|\mathcal C|\le D^n$ with $D$ independent of the alphabets would imply a constant-base sunflower bound after the standard random-coloring reduction.

Proposed replacement for spread refinement. Instead of maintaining a spread subfamily, maintain:

• per-coordinate collision information,
• a fixed labelled hash window for low-collision coordinates,
• terminal cylinder shadows for high-mass values,
• fresh/old/scale potentials that pay for repeated use.

Audit target. The attempt succeeds only if every nonterminal transition can be assigned to a finite list of local account inequalities, with constants independent of $n$ and independent of the alphabet sizes.

4) WORK

4.1. Standard reduction to balanced two-collision codes

Pad a rank-$n$ sunflower-free family to an $n$-uniform family by adding private dummy elements. Randomly color the ground set by colors $1,\ldots,n$ and keep the colorful sets. Some coloring keeps at least $(n!/n^n)|\mathcal F|$ sets. Encoding a colorful set by its unique element of each color gives a code \[ \mathcal C\subseteq \prod_i\Omega_i . \] If three codewords have no coordinate with exactly two equal symbols, then in each color class the corresponding set-elements are either all equal or all distinct, so the three pairwise intersections are equal and the original three sets form a sunflower. Thus the resulting code is balanced two-collision.

Therefore an alphabet-free estimate $|\mathcal C|\le D^n$ implies $|\mathcal F|\le (eD)^n$.

4.2. Terminal entropy descent

Let $\mu$ be the uniform law on a balanced two-collision code $\mathcal C$. The proof tree samples one or more independent codewords from $\mu$, but the terminal object is only a projected cylinder: \[ \theta=((J_s,a_s))_{s\in S}, \] meaning $X^s_{J_s}=a_s$ for each sample label $s$. Write \[ T(\theta)=\sum_s |J_s|,\qquad R(\theta)=|S|. \] If a terminal cylinder has mass \[ \Pr[\theta]\ge \exp(-\lambda T(\theta)-cR(\theta)), \] then the usual induction on rank gives $|\mathcal C|\le D^n$ whenever $\log D\ge \lambda+c$. Indeed, each cylinder $\{x:x_{J_s}=a_s\}$ is again a balanced two-collision code after deleting the fixed coordinates.

Thus the goal is to produce a random terminal projected cylinder $\Theta$ with \[ \mathbf H(\Theta) \le \mathbb E[\lambda T(\Theta)+cR(\Theta)] \] for absolute constants $\lambda,c$.

4.3. Fixed labelled hash window

For a one-coordinate law $p$, write \[ \operatorname{Coll}(p)=\sum_a p(a)^2. \] If $X,Y,Z$ are independent samples and coordinate $i$ has collision at most $\beta$, then \[ \Pr[i\in \operatorname{Split}(X,Y,Z)]\le 3\beta. \] Hence a block $H$ of $h$ low-collision coordinates satisfies \[ \Pr\left[|\operatorname{Split}_H(X,Y,Z)|\le 6\beta h\right]\ge 1/2. \]

The proposed mechanism collects low-collision coordinates in a fixed labelled hash window. The label is one of the active sample labels. The window has a fixed transcript budget $B_{\rm win}$ and stops before exceeding that budget. High-mass values are not exposed recursively: if a coordinate develops an atom of mass at least $\beta$, the branch terminalizes by pinning that value as a projected shadow.

If a nonterminal labelled window obtains $h$ surviving coordinates, three conditioned copies under the same public transcript produce a near-sunflower block. The transcript key has Renyi-3 copy cost \[ 2\mathbf H_3(K)+O(1)\le 2\mathbf H(K)+O(1)\le O(B_{\rm win}). \] The sparse split exposure costs $O(h)$. Thus the local hash-return cost is \[ C_{\rm hash}=O(B_{\rm win}+h), \] with $B_{\rm win}$ and $h$ fixed constants.

4.4. Product-KL routing

In a close-pair node, one has a block $J$ on which two sampled codewords agree and the third differs: \[ X_J=Y_J=A_J,\qquad Z_J=B_J\ne A_J . \] The intended product-KL branch theorem scans the actual tuple-coordinate list \[ (A_{j_1},B_{j_1},A_{j_2},B_{j_2},\ldots). \] Every branch must route to exactly one of:

1. an amortized terminal prefix;
2. a one-coordinate heavy value pin;
3. a charged pruning event not inherited by descendants;
4. a genuine low-collision law of an actual sample coordinate $X_j$, $Y_j$, or $Z_j$, sent to the labelled hash window.

The critical requirement is that the low-collision law is the actual conditional sample-coordinate law, not a law on the pair alphabet $(A_j,B_j)$ and not the auxiliary reference product law.

4.5. Global account inequality

The fixed hash window gives only a one-time near-sunflower block. It does not produce a reusable spread subfamily. The proposed replacement invariant is a state with potential \[ \Phi = B_f |[n]\setminus U| +(B_o|U|-N_{\rm old}) +B_s\sum_{\text{active }J}|J|. \] Here $U$ is the old support, $N_{\rm old}$ is a weighted count of registered old incidences, and the last term pays active scale/rank bookkeeping.

The desired branch inequality is \[ \operatorname{spent}(\pi) + C_{\rm hash}q(\pi) + \Phi(S_\pi) \le \Phi(S_0)+\operatorname{terminal\ shadow\ charge}(\pi), \] where $q(\pi)$ is the number of completed hash windows on the branch.

At the one-step level, the required inequalities have the form \[ (B_f-B_o)|F| \ge \text{fresh selector cost}+\text{child deposit}+\text{possible hash token}, \] \[ \omega_{\rm old}m \ge \text{old transcript cost}+\text{child deposit}+\text{possible hash token}, \] and \[ \text{rank/scale drop} \ge \text{new active-block deposit}+\text{selector cost}. \] If all transition modes are exhaustive and these inequalities cover every selector, the entropy criterion above would close with constants independent of the alphabet.

5) VERIFICATION / ATTEMPTED BREAKS

Known verification obligations. The following are the main places where the attempt can fail:

1. The fixed-window hash return must really have per-window cost $O(B_{\rm win}+h)$, not a hidden per-branch or alphabet-dependent cost.
2. The product-KL branch theorem must output actual sample-coordinate laws.
3. Completed hash-window tokens must be matched injectively to hard exits; tokens cannot pile up along a nested chain.
4. The residual repair procedure must immediately route every returned near-sunflower block to fresh, old, scale, or terminal progress.
5. The transition system must be exhaustive; no branch may continue without a terminal shadow or a potential drop.

Falsification tests reported by the contributor. The contributor reports using executable smoke tests for the following checks. The supporting scripts are not included in this submission:

• alphabet stretching of toy balanced two-collision codes;
• multi-label hash-token matching;
• terminal cylinder potential compatibility;
• transition-exhaustiveness signatures;
• constant-hierarchy acyclicity;
• old-window queue refinement;
• hash-prefix budget handling;
• Renyi copy-tax accounting;
• collision-to-split estimates;
• high-revisit branch exhaustiveness;
• randomized ledger walks.

These tests are not formal proofs. They are only attempts to catch common bookkeeping failures.

Internal LLM audit status. Several long GPT Pro, Claude Opus, and Codex adversarial-checking passes reported that the current interface-level ledger appears closed and did not identify a remaining bookkeeping counterexample. This is recorded only as provenance for the attempt. It is not expert verification, not a formal proof, and not a claim that the sunflower conjecture is solved.

6) FINAL

PARTIAL / UNRESOLVED.

This is not a complete proof of the Erdos--Rado sunflower conjecture. The main contribution of the attempt is a proposed conditional framework: if the product-KL routing, fixed-window hash return, residual repair, and global account inequality are all correct with alphabet-free constants, then the usual terminal-cylinder entropy descent would yield a constant-base bound for balanced two-collision codes, and hence for three-petal sunflower-free families after the standard reduction.

The exact open gap is: \[ \text{prove, without ``by inspection'', that every transition branch is covered by the one-step account inequalities above.} \]

7) COMPLETION ESTIMATE

COMPLETION ESTIMATE: 20%.

This percentage reflects that the approach has a defined reduction, candidate local mechanisms, and executable falsification tests, but still lacks an independent verification of the full transition-system exhaustiveness and global charge inequality.

8) DISCLOSURE / PROVENANCE

This file is not the transcript of a single model run. It is a Codex-organized summary of a much longer LLM-assisted research attempt. The underlying framework was developed through many extended conversations with frontier LLMs, including GPT Pro, Claude Opus, and Codex. Some of those conversations became too long to open reliably, some are no longer accessible in full, and the Codex continuation itself used compressed conversation summaries. Consequently this submission should be read as a reconstructed and organized record of the LLM attempt, not as a verbatim complete transcript.

The contributor also prepared a longer draft, a short roadmap, audit notes, and smoke-test scripts as separate research materials. Those materials are not included in this submission. No part of this submission should be treated as expert-verified.

9) REFERENCES

References

• [ALWZ] R. Alweiss, S. Lovett, K. Wu, and J. Zhang, Improved bounds for the sunflower lemma, Ann. of Math. $2$ 194 $2021$, no. 3, 795--815.
• [Rao] A. Rao, Coding for sunflowers, Discrete Anal. $2020$, Paper No. 2.
• [Tao] T. Tao, The sunflower lemma via Shannon entropy, blog post, 2020.
• [BCW] T. Bell, S. Chueluecha, and L. Warnke, Note on sunflowers, Discrete Math. 344 $2021$, no. 7, 112367.
• [ER] P. Erdos and R. Rado, Intersection theorems for systems of sets, J. London Math. Soc. 35 $1960$, 85--90.