Spread families whose sunflowers all have large cores

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_proof: Powers of the triangle are intersecting and sunflower-free with an exactly computable excess; substituting them into block-family powers makes every sunflower core a union of whole inner members, while the block levels absorb the excess.

evidence/: Exact checks of the triangle powers, the core lemma on two small substitutions and the spread recursion behind L215.

This result is recorded by Plasma AI. It sharpens L214, which shows that spread does not force two disjoint members, in the direction that matters for Problem 20: spread does not force a sunflower with a small core either. It does not settle Problem 20, and its families do contain $k$-sunflowers for every $k\le\rho$. The proof gives the core lemma and the spread computation. The proof sketch for Problem 20 records what this leaves for an attack.

Significance

Problem 20 is equivalent to the statement that, for each $k\ge3$, families that are spread enough contain a $k$-sunflower. That equivalence is the link argument recorded in the research. L214 shows that the sunflower cannot always have an empty core. This claim shows more: for every spread $R$ and every bound $s$, some intersecting family with $R$-spread uniform measure has no $k$-sunflower with a core of fewer than $s$ points. So no statement of the form "spread at least $R_k$ forces a $k$-sunflower whose core has at most $s_k$ points", with $R_k$ and $s_k$ independent of the set size, can hold. Along these families, at fixed $R>3$, every core has at least about $\log_2\log_2n/\log_2(R/3)$ points. Cores of exactly $2^g$ points occur, so the bound is attained in each family.

The mechanism is a substitution of the powers of the triangle, which are intersecting and contain no $3$-sunflower, into the block-family powers of L214. Every sunflower of the result has its core made of whole members of the inner family, while the outer levels halve the excess that the inner family carries.

Dependency roles. From L214 this claim uses the definitions in its statement and, in the proof, Lemma 1 (substitution keeps a family uniform and intersecting), Lemma 2 (the exact spread recursion $E_{G[H]}(R)=\psi_G(E_H(R))$) and Lemma 3 (the block-family bound, which gives $\psi_B(E)\le E/2$ for $E\le\rho$ when $m=\rho^\rho$), all from the L214 proof.

Standing and verification

Tier 0, author-recorded. The proof is complete, elementary and deterministic. No independent review and no formalization exist. The claim's statement is in the normalized convention. The separate thinning argument without a family-size cap gives the corresponding absolute-spread obstruction at author standing; it is a consequence recorded in research, not an additional clause of this claim's statement.

The evidence program checks, by exhaustive or core-by-core search in exact arithmetic, that $D$, $D^2$ and $D^3$ are intersecting and free of $3$- and $4$-sunflowers, and that the excess of $D$ and $D^2$ matches Lemma A at three rates. It checks Lemma B on $B(3,3)[D]$ and $B(2,3)[D]$: both are intersecting and free of $4$-sunflowers, and their smallest $3$-sunflower cores have exactly $2$ and $4$ points. These are the predicted values, since every $3$-sunflower of $B(2,3)$ has a core of two points. It also checks the spread recursion on $B(3,3)[D]$ at five rates. It does not check the statement for general $g$ and $R$. From the repository root, run

uv run --no-sync python wiki/theory/set_systems/L215_spread_large_core_sunflowers/evidence/main.py

which takes a few seconds and exits with status 0 when every check passes.

Nothing remains open within the stated scope.