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Target and standing

The target is Problem 20, which the catalog records as open; nothing here changes that. Let $f(n,k)$ be the least number such that every family of at least $f(n,k)$ distinct $n$-element sets contains a $k$-sunflower: $k$ distinct members whose pairwise intersections all equal one set, the core, which may be empty. The question is whether $f(n,k)<c_k^n$ for each fixed $k\ge3$, with $c_k$ independent of $n$ and of the ground set. The case $k=2$ is immediate. A disproof needs one $k$ and $k$-sunflower-free families of $n$-sets whose size to the power $1/n$ is unbounded; a bound exponential in the size of the ground set does not suffice when that size is unrestricted. The general upper bound used for comparison here, for $n\ge2$, is $f(n,k)\le(Ck\log n)^n$, from Bell–Chueluecha–Warnke's Theorem 1.

Problem 20 is not solved. Every result recorded for this attack is author-recorded by Plasma AI and carries at most tier 0. The claims L214 and L215 record the spread obstructions; the structured positive theorem is L268. An outside preprint, Fukuyama's arXiv:2212.13609v3, announces a proof of the conjecture with base $Ck^4$. The explicit product-family test contradicts its Lemma 2.1 and shows that its core-selection inequality (2.4) need not have any solution under the stated hypotheses. This is an author-recorded failure of intermediate assertions, not a disproof of the conjecture. The preprint is not used as a premise.

Conventions

A family of $n$-sets is absolutely $R$-spread if every nonempty $T$ lies in at most $R^{n-|T|}$ members (the convention of Bell–Chueluecha–Warnke and of Rao), and normalized $R$-spread if every nonempty $T$ lies in at most $R^{-|T|}|\mathcal F|$ members, which forces $|\mathcal F|\ge R^n$ (the convention of Alweiss, Lovett, Wu and Zhang and of Tao). Normalized $R$-spread implies normalized $R'$-spread for $R'\le R$. The thinning lemma converts normalized $Q$-spread to an absolute $r$-spread subfamily larger than $r^n$ when $r\ge2$, $n\ge2$ and $Q\ge4er(8r)^{2/(r-1)}$; $Q\ge32r$ suffices for $r\ge16$. It has no upper bound on the original family size. The narrower thinning statement in L214 remains sufficient for that claim.

Held links

1. Link reduction (elementary, author-recorded). If $|\mathcal F|>R^n$, choose an inclusion-maximal $T$ with $|\mathcal F_T|>R^{n-|T|}$. The empty set qualifies and $|T|<n$. The link $\{S\setminus T:S\in\mathcal F_T\}$ is absolutely $R$-spread with more than $R^{n-|T|}$ members, and a $k$-sunflower in it lifts to a $k$-sunflower of $\mathcal F$ whose core contains $T$. The argument is spelled out in the obstruction note.

2. Spread form (equivalence, elementary). By link 1, Problem 20 holds, after a harmless enlargement of the base to handle strict endpoints, if

   SF($k$): every absolutely $R_k$-spread family of $n$-sets with more than $R_k^n$ members contains a $k$-sunflower, for every $n$.

   Conversely Problem 20 with constant $c_k$ gives SF($k$) for every $R_k\ge c_k$, since such a family has more than $c_k^n$ members. In the forward direction SF($k$) gives $f(n,k)\le\lfloor R_k^n\rfloor+1$, and, for example, $c_k=R_k+2$ meets the target's strict inequality. Thus SF($k$) is equivalent to the target; it changes the vocabulary and supplies the spread hypothesis, but not the difficulty.

3. The matching route is closed. The sufficient condition $M(k)$, that absolutely $R_k$-spread families with more than $R_k^n$ members contain $k$ pairwise disjoint members, is false for every $k\ge3$. For $R_k\ge16$, L214 supplies the counterexamples. The general thinning argument above supplies them for every $R_k\ge2$; when $R_k<k-1$, a family of $k-1$ singletons already refutes the assertion. For $k=2$, every $R_k\ge2$ fails, but $R_k=1$ must be excluded: absolute $1$-spread forces disjointness. These distinctions do not affect the target $k\ge3$. This also refutes the proposed asymptotic strengthening beside Rao's Lemma 2 that the lemma might hold with $r(p,k)=O(p)$. Varying the substitution scales in the refined obstruction gives a lower obstruction of order $\log n/\log\log n$ along its uniformities. This normalized order was already recorded by ALWZ, Section 4 after Theorem 4.2, with attribution to Lovett–Solomon–Zhang (2019); no new asymptotic lower bound is claimed. The upper bound $C\log n$ is Bell–Chueluecha–Warnke's Lemma 2, in the absolute convention. The normalized comparison is stated with its source qualifications in the obstruction note.

4. Two obstructions at their scope. The product of $n$ blocks of size $s$ obstructs the random-hitting lemma at constant spread but has $s$ disjoint members (finite-set threshold note). The families of L214 obstruct the matching lemma and are intersecting, so every sunflower they contain has a nonempty core. A proof of SF($k$) must therefore find sunflowers with nonempty cores in some spread families.

5. Cores must be allowed to grow. L215, which substitutes sunflower-free triangle powers into the block families gives, for every $R\ge3$ and every $s$, an intersecting family with $R$-spread uniform measure in which every $k$-sunflower, $k\ge3$, has a core of at least $s$ points. So no lemma "spread at least $R_k$ forces a $k$-sunflower with a core of at most $s_k$ points", with $R_k$ and $s_k$ independent of $n$, holds. At fixed $R>3$ the forced cores have about $\log_2\log_2n/\log_2(R/3)$ points. The general thinning lemma gives the corresponding obstruction for absolute $r$-spread families with more than $r^n$ members, for every $r\ge2$.

6. Threshold for three petals. Powers of a ten-triple intersecting seed on six points, recorded in the obstruction note, are intersecting, have no $3$-sunflower, and have normalized spread approaching $\sqrt{10}$. So $R_3\ge\sqrt{10}$ in the normalized form of SF($3$), above the value $2$ of the Erdős–Rado product.

7. A maximal-rate chain need not find the core. The chain obstruction gives, for every $A>5$, an $n$-uniform family larger than $A^n$ with matching number two, whose unique maximal-rate choices lead to a three-sunflower-free terminal family. An explicit sunflower is discarded along the way. Its proof uses the thinning lemma, the triangle powers, and an elementary union-and-cone recurrence, all at author standing. Favorable tie-breaking does not repair the procedure.

8. Affine linear words satisfy an exponential bound. L268 proves that a linear subspace $C\le A_1\oplus\cdots\oplus A_n$ over $\mathbb F_q$, of dimension at least $sn$, contains an $s$-dimensional subspace on which every coordinate is zero or injective. Its $q^s$ words form a sunflower. Thus affine linear word families without a $k$-sunflower have fewer than $k^{2n}$ members, uniformly over the field and coordinate alphabet dimensions. The coset argument gives an exponential bound for dense subsets of such a linear family, with dependence on the density. The proof includes the independent-representative lemma it uses. This structured result has author standing and no established novelty claim.

9. Bounded additive growth also suffices. In the binary encoding, the covering argument shows that a sunflower-free family with growth ratios $K_2=|2F|/|F|$ and $K_4=|4F|/|F|$ has $|F|<(2(k-1)K_2 2^{K_4})^n$. Thus fixed fourfold growth gives an exponential bound. With the explicitly cited Plünnecke estimate, doubling at most $K$ gives base $2(k-1)K2^{K^4}$. This extends the positive result beyond linear families, but arbitrary words need not satisfy the growth hypothesis. An explicit binary Sidon family shows that extracting a bounded-doubling subfamily can require exponential loss even when the original family is sunflower-free.

10. Links on few points, sections, and an intersecting normal form. The slice-rank note records, for $k=3$, a routine projection form of Naslund–Sawin's slice-rank bound for words and its consequence for set families: more than $\rho^m$ sets of size $m$ on at most $A\rho m$ points contain a $3$-sunflower once $\rho\ge e^3(27/4)A^2$, about $135.6A^2$, whatever the cores. So every link that link 1 produces from a counterexample at base $\rho$ covers more than $m\rho^{3/2}/11.65$ points. The note also proves that the target for $k=3$ is equivalent to the statement that, for some $c>1$, intersecting, absolutely $c$-spread families of $j$-sets without a $3$-sunflower have at most $(c-1)^j$ members. The slice-rank bound finds $3$-sunflowers in the families behind the Alweiss–Lovett–Wu–Zhang lower bound for robust sunflowers, and sections, whose sunflowers lift exactly, reach the unthinned families of L214 and L215. For the L214 thinning, a matching in a projected link finds a sunflower with a one-part core. The displayed section count tests can simultaneously fail after thinning, even with adaptive section choice; other links and sections are not excluded. These arguments are author-recorded; no novelty is claimed for the slice-rank inputs.

11. Counting in affine spaces and a Fourier criterion. The agreement-statistics note extends the Hall selection in the proof of L268: $k$ independent uniform words in any binary affine word space satisfy the coordinate sunflower pattern, allowing the all-equal tuple, with probability at least $2^{-(k-1)^2n}$. The homogeneous indicator has Fourier norm at most $(k!+1)^n$, including after restriction to the affine space. Therefore a family $F$ in a binary affine space $a+C$ contains a $k$-sunflower if $|F|>2^{(k-1)n+1/(k-1)}$ and its uniform measure has squared bias at most $1/(2((k!+1)2^{(k-1)^2})^n)$ on every nontrivial character of $C$. For $k=3$ the thresholds are $\sqrt2\,4^n$ and $1/(2\cdot112^n)$. The bias condition is not deduced from cardinality or spread. The note also proves that uniformly sampled families of $\lfloor\rho^n\rfloor$ words in $[B]^n$ have a $k$-matching with probability tending to one precisely above the exponential rate $\gamma_k(B)=B/((B)_k)^{1/k}$, for fixed $B\ge k$, away from equality. These are author-recorded statements. They neither bound the largest adversarial family nor confer independent standing.

The open gap

The whole difficulty is SF($k$) itself: a way to produce a $k$-sunflower from spread alone, with a nonempty core when the family has no $k$ disjoint members, and with a core that may need to grow like $\log\log n$.

Where the core must be found (author-recorded observation). For a family $\mathcal F$ of $m$-sets and a rate $R$, call $\Pr[T\subseteq S]\,R^{|T|}$ the weight of a nonempty set $T$; a member has weight $R^m/|\mathcal F|$. If a set of largest weight among all nonempty sets is not a member, its link is again normalized $R$-spread, because no superset is heavier. Every member has largest weight exactly when $|\mathcal F_T|\le R^{m-|T|}$ for all nonempty $T$, that is, when $\mathcal F$ is absolutely $R$-spread. So following the heaviest sets, a form of the link reduction of link 1, moves to a spread link while some proper subset outweighs the members. It stops exactly at absolutely spread families, where no proper subset outweighs a member. There the sunflower has to be found with a light core, a set $T$ with $|\mathcal F_T|\le R^{m-|T|}$. The families of L214 keep their properties in the absolute form, by L214's Lemma 4, so such a light core must sometimes be nonempty. L215 and the general thinning lemma show that it must sometimes be large in either spread convention.

No general candidate below is proved. Each is a statement to test against three kinds of family. The L214 families have one-point cores. The L215 families have only large cores. The powers of the ten-triple seed have no $3$-sunflower and spread just below $\sqrt{10}$.

• Core-finding through light links, recursively. In an intersecting family every member meets a fixed member $S_0$, so some point $v\in S_0$ lies in at least a fraction $1/n$ of the members, and passing to such links loses a factor up to $n/R$ at every step, the loss behind the Erdős–Rado bound. A heavy point does not exist in general: by the remark on light core points in the obstruction note, the L214 families with more levels are still $\rho$-spread while every point lies in at most a fraction $2^{-h}$ of the members. There the sunflower's core point is light, but its link, with its own uniform measure, is again $\rho$-spread. A useful lemma would find a possibly light point or set whose link is again spread relative to its own size, and would continue in that link. By L215 it cannot always use a bounded-size core; a single step with a large core remains possible. The degenerate case is the one isolated above, an absolutely spread family in which no proper subset is relatively heavy.
• Structure of spread families without $k$ disjoint members. The proposition of the obstruction note shows that substitution powers behave like their seeds. A stability statement would reduce SF($k$) to the links. It would say that every spread family without $k$ disjoint members decomposes, like a seed, into a few heavy points and spread links.
• Words over unbounded alphabets. By Alon–Shpilka–Umans' Theorem 2.6, the target is equivalent, up to a factor $e$ in the base, to a bound for words over an arbitrary alphabet. The slice-rank bound of Naslund–Sawin's Theorem 5 grows with the alphabet; its projection form in link 10 depends on the family's projections to two thirds of the coordinates instead, but a family whose such projections are nearly injective escapes it. The linear subclass in link 8 admits a bound independent of the alphabet without compressing it. A general family need not contain the linear structures supplied by that argument; no reduction of general words to the linear subclass is given here.
• A dichotomy for unbalanced links. By link 10, a counterexample to SF($3$) at rate $\rho\ge137$ yields an intersecting, absolutely $\rho$-spread family of $m$-sets with more than $(\rho-1)^m$ members on more than $m(\rho-1)^{3/2}/11.65$ points, with no section certificate in any link. A proof could split such families into a pseudorandom case, where counting in some link gives three pairwise disjoint members uniformly in the alphabet, and a structured case, where a passage to a link, section or projection gains in an explicit invariant. Link 11 gives concrete sufficient Fourier conditions for the counting case, also allowing arbitrary cores, but only an exponentially small character bias when they fail. Converting that bias into a useful restriction with controlled total loss is unproved. Any such statement must handle random families, the unthinned and thinned substitution powers, and the two-part chain obstruction of link 7 at once. No such dichotomy is proved; since a sunflower is a matching in the link at its core, stating the residual this way is a reformulation of SF($3$), not a reduction of it.

Source interfaces and their standing

• Bell–Chueluecha–Warnke, arXiv:2009.09327v2: Theorem 1, Lemma 2 and the quoted Theorem 3 of Rao and Tao, read on the library cards; their proofs are source readings, not independently reviewed here. The comparison in link 3 uses Lemma 2 as stated, in the absolute convention with at least $r^n$ members.
• Rao, Coding for sunflowers, arXiv:1909.04774v2: the primary proof, including the encoding argument in Section 4, was read. Lemma 2 and the remark on p. 2 are used at source standing. The remark is refuted only as far as the explicit constructions state: L214 covers $r\ge16$, and the general thinning lemma extends the counterexamples to $r\ge2$.
• Alweiss–Lovett–Wu–Zhang, arXiv:1908.08483v3: Theorem 2.5 in the weighted convention, used only for the normalized upper bound in the obstruction note; Rao's sharper bound is used only as they record it. Section 4's Theorem 4.2, its proof, and the immediately following comparison to Lovett–Solomon–Zhang were read in the primary transcription. The original cited lower-bound construction has not been checked here. Lemma 3.1, the lower bound for robust sunflowers, and its proof were read in the primary transcription, with the statement and the start of the construction checked on the page images of pp. 11–12. This construction supplies the comparison family tested in link 10; the general slice-rank estimate does not depend on it.
• Fukuyama, arXiv:2212.13609v3: Section 2's four-step proof was read, including the definitions and the interfaces from Lemma 2.1 through Proposition 2.3. The exact source test and page-image receipt are in the linked gap analysis. This is a failed-argument comparison, not a premise or an accepted solution.
• Alon–Shpilka–Umans: Definition 2.5 and Theorem 2.6, including its proof, were read in the primary transcription, pp. 5–6, with the relevant definition checked on the p. 5 image. The theorem supplies the word formulation comparison, and its colouring step is reused in link 10. Naslund–Sawin's arXiv:1606.09575v1, Lemma 6 and the Section 3 proof of Theorem 5 were read in the primary transcription. Lemma 6's statement was also checked on the p. 2 image; Section 3 was read as text. Link 10 proves the bound it uses in full, with one external input at source standing: Tao's lemma that a diagonal tensor with nonzero diagonal entries has slice rank equal to the size of its support, as stated in Naslund and Sawin's Lemma 6. Link 11 uses elementary Fourier orthogonality and the proved linear representative lemma from L268; no additional external theorem is its premise.
• Petridis, arXiv:1101.3507v3: Theorem 1.1 at $A=B$, $h=4$, supplies $|4A|\le K^4|A|$ from $|2A|\le K|A|$. Proposition 2.1 and Theorem 3.1 were read as primary proof text, without page-image inspection. This premise is used only in the doubling corollary of link 9. The fourfold version is self-contained given link 8.

Maximal-rate links: exact successes and a general obstruction

A canonical chain (author-recorded). For a family $\mathcal F$ of $m$-sets, $m\ge2$, let

$$R_a(\mathcal F)=\max_{0<|T|<m}|\mathcal F_T|^{1/(m-|T|)},$$

the least rate at which $\mathcal F$ is absolutely spread, and let $T^*$ attain it. The link at $T^*$ has exactly $R_a^{m-|T^*|}$ members and is absolutely $R_a$-spread: for nonempty $U$ disjoint from $T^*$, $|\mathcal F_{T^*\cup U}|\le R_a^{m-|T^*|-|U|}$, by the choice of $R_a$ when $|T^*\cup U|<m$ and trivially otherwise. Hence $R_a$ of the link is at most $R_a(\mathcal F)$. Iterating gives a chain of links whose rates never increase. At each step either the current link has $k$ disjoint members, which gives a $k$-sunflower whose core is the union of the sets $T^*$ chosen so far, or the chain continues. When a step reaches uniformity $1$, its size is the preceding selected rate, and its members are distinct singletons. So the chain ends in a $k$-sunflower unless its rate falls below $k$. Without $k$ disjoint members, a cover of at most $(k-1)m$ points gives a point of degree at least $|\mathcal F|/((k-1)m)$. That bounds the loss of rate per step by the factor $((k-1)m/R)^{1/(m-1)}$, the loss behind the Erdős–Rado bound.

The exact calculation on L214 and L215 shows that a whole inner member over the first center of the top block seed attains $R_a$, and that its link has $\rho$ disjoint members. Hence for $k\le\rho$ a suitable maximizer succeeds at the first step on these families. This does not prove that every maximizer succeeds.

The general decay bound is false, even when the maximizing set is unique at every step before a sunflower-free family is reached. The full construction starts with a low-rate sunflower-free family $G$. It forms a new family by adjoining one new point to every member of $G$, then adding an intersecting absolutely spread family on a disjoint ground set. The selected link always returns $G$. The added family nevertheless increases the size enough that iteration makes the initial exponential rate arbitrarily large. All visited families have matching number two. One added part can contain an explicit three-sunflower that the chain misses.

Remaining obligation

The equivalent assertion SF($k$) is still open. A new core-finding argument must handle both the selected link and the members it discards, or use a different invariant. Merely bounding the density loss along a single maximal-rate chain cannot work. The obstruction uses two intersecting parts at each stage, so a recursion that retains the other part remains a possible route, but it needs a bound on the combined recursion. No such bound is established here.

One precise question is whether the absence of a $k$-sunflower bounds the total mass of a tree of links, even though the largest individual link can lose all useful structure. Any proposed weight must accommodate disjoint unions and coning, as in the chain obstruction, as well as substitution, as in L214 and L215. The algebraic result in link 8 supplies a uniform dimension bound for linear words, and its coset partition also handles subsets of fixed positive density in a linear family. Extending that argument requires a structural reduction that retains words of the original family. No such reduction or general alphabet compression is proved here. One sufficient, unproved extraction statement is the following: for each fixed $k$, constants $B_k,K_k$ exist such that every $k$-sunflower-free word family $F$ of length $n$ has a subfamily $F'$ of at least $|F|/B_k^n$ members, with injective binary vector encodings of its coordinate alphabets satisfying $|F'+F'|\le K_k|F'|$. Link 9 would then bound word families with exponential base $2B_k(k-1)K_k2^{K_k^4}$. The Alon–Shpilka–Umans reduction costs a further factor $e$ for arbitrary set families, with harmless endpoint slack. No such extraction theorem is established. In fact, with no further requirements on $B_k,K_k$ or the extraction, this statement is equivalent to the target: an exponential bound for sunflower-free word families permits taking $F'$ to be a singleton, with $K_k=1$ and $B_k$ at least the bound's base. The proposed extraction is therefore a reformulation, not a reduction to a known weaker problem. A useful advance along it would need a structural extraction argument independent of the desired cardinality bound. The binary Sidon example rules out a loss that is always subexponential, or always retains a fixed positive power of the original cardinality. It does not rule out the exponential loss that would suffice. Establishing such an extraction in the sunflower-free case, or using large growth in a different way, remains unproved.

Link 10 narrows where a counterexample at rate $\rho\ge137$ for $k=3$ can live: an intersecting trace uses more points than its uniformity times $(\rho-1)^{3/2}/11.65$, with no section certificate in any link. The random thinnings of substitution powers considered there contain a sunflower with the explicit high probabilities in Corollary 9. Link 11 proves that random word families at sufficiently large constant rates have matchings with probability tending to one. A candidate must have correlations that evade the recorded section, projection and counting criteria.

The core-allowing Fourier criterion in link 11 supplies a more concrete inverse question. A sunflower-free family above its size threshold has nontrivial Fourier bias, in every binary encoding and relative to its affine span. The guaranteed bias is exponentially small in $n$. Selecting the larger fiber of one such character can lose almost half the words and need not fix a coordinate. There is no bound here on the number or cumulative cost of those restrictions, and the affine dimension is unrestricted. A useful inverse must retain original words, control that total cost in terms of $n$, and handle the discarded mass when necessary. The dichotomy described under the open gap would do this. It is not established.