Sunflower exponential bound

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additive_word_families: Binary word families with bounded fourfold growth lie densely in an affine linear family and obey an exponential sunflower bound. Standard sumset estimates give a bounded-doubling corollary.

agreement_statistics: A counting form of linear Hall bounds the probability of the sunflower coordinate pattern in binary affine word spaces, including all-equal tuples. Small Fourier bias forces genuine sunflowers above an exponential size threshold. Random-matching thresholds and equal pair profiles test this route; the structural inverse remains open.

fukuyama_sequence_gap: Full transversals satisfy the spread hypothesis of arXiv:2212.13609v3 but violate Lemma 2.1 and admit no sequence satisfying (2.4). The missing factor in Case 3 cannot be supplied by the stated hypotheses.

linear_word_families: A linear Hall argument finds a subspace on which every coordinate map is zero or injective. It gives an alphabet-independent exponential sunflower bound for affine linear word families over finite fields.

maximal_rate_chain_obstruction: For arbitrarily large exponential size, a family with matching number two can force every maximal-rate link chain into a sunflower-free family. An explicit sunflower can be retained outside the chosen chain.

maximal_rate_links: Exact excess formulas show that the maximal-rate link procedure succeeds at its first step on L214 and L215, although these families defeat matching and bounded-core statements at fixed spread.

proof_sketch: Problem 20 is equivalent to a sunflower statement for spread families; L214 refutes the constant-spread matching route and L215 any bounded-core variant. Maximal-rate link chains can fail; a linear Hall argument bounds affine linear word families, and slice rank handles links on few points. A counting extension gives Fourier criteria allowing arbitrary cores. The general structural inverse remains open.

refined_spread_obstructions: Decreasing block scales give intersecting families with spread of order log n/log log n. An anchored first-moment argument converts normalized spread to absolute spread without a bound on the original family size.

slice_rank_spread_links: The slice-rank bound for words, in projection form, gives three-petal sunflowers in families of m-sets on few points, so every spread link in a counterexample is unbalanced. Sections reach the unthinned obstruction families, and projected matchings reach their random thinnings. The general dichotomy remains open.

spread_matching_obstruction: Substitution powers of intersecting seeds have an exactly computable spread whose limit is a one-level seed quantity; block seeds give intersecting families with spread log n/(log log n)^2, and inner sunflower-free seeds force large sunflower cores.

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This folder holds the attack on Problem 20, the Erdős–Rado sunflower problem through spread families. The proof sketch states the target, the equivalent spread form, the links held so far at their standing, and the open gap. The obstruction note explains the substitution construction behind the claims L214, which gives intersecting families at arbitrarily large spread, and L215, which shows that spread does not force a sunflower with a bounded core, and records what that leaves for a proof.

The existence of highly spread intersecting families is already known from the literature cited on L214; the substitution formulas here provide an explicit alternative. The maximal-rate chain obstruction shows that always choosing a densest link can miss a sunflower even at arbitrarily large exponential size. The product-family source test identifies a specific failed intermediate assertion in an outside claimed proof. These are author-recorded arguments. They do not settle the full question or establish acceptance of the pending claim on the problem page.

On the positive side, the linear Hall argument of L268 proves an exponential bound for affine linear word families over arbitrary finite fields and arbitrarily large coordinate alphabets. Its dense-subset corollary also bounds families of fixed positive density in an affine linear family. The additive-growth extension also handles nonlinear binary word families of bounded doubling or fourfold growth. Their exact remaining gap is a structural argument for general word families; passing to a linear span can introduce words that were not in the family.

The slice-rank note treats three petals whatever the core. A routine form of the slice-rank bound gives sunflowers in families of $m$-sets on few points, so every spread link in a counterexample uses many more points than its spread forces. The note also reduces the three-petal case to intersecting spread families, and records which obstruction families the bound reaches through sections and projected links, and which remain.

The agreement-statistics note extends the linear Hall argument to a uniform lower bound on the probability of the coordinate sunflower pattern in binary affine spaces, including all-equal tuples. Subtracting those tuples and using an explicit Fourier norm bound gives a criterion for genuine sunflowers with arbitrary cores. The note also proves the exponential threshold for matchings in random word families and gives two families with identical aggregate pair-agreement profiles but different sunflower behavior. A structural argument turning Fourier bias into a restriction with controlled total loss is still missing.