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CS-765 · computer science · mathematics · Graph theory and combinatorics

Spreadness as a Replacement for Sunflower Structure

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Statement

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

A sunflower is a family of sets whose pairwise intersections all equal the same core. Spreadness bounds the fraction of the family containing each prescribed subset, with the bound depending on its size. Robust variants allow the required petals to appear with a stated probability.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The application family, spread parameter, and required implication are not fixed by the supplied direction.

Source-record limitation: No particular sunflower application, spreadness parameters or replacement theorem is specified; this is an open-ended methodological question.

Statement recorded in the supplied source.
Can spreadness replace sunflower structure in theoretical-computer-science applications that currently use sunflowers, particularly through robust-sunflower generalizations? [S2]
Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Replace sunflower structure by quantitative spread conditions in the stated applications.

Sources

Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30005829 and code OWR-14298166-005 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

CS-765-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on CS-765

Spreadness as a Replacement for Sunflower Structure

Definitions and mathematical statement

Shared notation.
Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

A sunflower is a family of sets whose pairwise intersections all equal the same core. Spreadness bounds the fraction of the family containing each prescribed subset, with the bound depending on its size. Robust variants allow the required petals to appear with a stated probability.

Record scope and qualifications.
The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The application family, spread parameter, and required implication are not fixed by the supplied direction.

Source-record limitation: No particular sunflower application, spreadness parameters or replacement theorem is specified; this is an open-ended methodological question.

Statement recorded in the supplied source.
Can spreadness replace sunflower structure in theoretical-computer-science applications that currently use sunflowers, particularly through robust-sunflower generalizations? [S2]
Source status.
The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Replace sunflower structure by quantitative spread conditions in the stated applications.

Sources

Preparation and review.
The original source and UnsolvedMath contributors retain attribution for the source extract. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, scope notes and formatting. No new human mathematical authorship or independently verified solution is asserted. Permanent source ID 30005829 and code OWR-14298166-005 checked against the complete supplied registry and current repository. Distinct upstream records, including repeated questions and broad or damaged extracts, are retained for complete coverage of the source CS category. A repeated record is not a new independent conjecture. Complete mathematical-equivalence and current-literature review has not been performed. Preparation and final review follow contribution prompts C and D, with source gaps disclosed. Structural and rendering checks do not establish a new theorem or open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

A second-moment route for a precisely defined spread model

The workshop discusses several sunflower variants and applications. Rather than silently repair the direction of containment in an extracted robustness definition, we specify our own auxiliary random-containment experiment. Let $\mathcal F$ be a nonempty family of $k$-element subsets of a finite universe, $k\ge1$. Suppose a uniform $B\in\mathcal F$ satisfies \[ \Pr(S\subseteq B)\le r^{-|S|}\quad\hbox{for every subset }S, \] where $r>0$. Let $R$ include each universe element independently with probability $0<p\le1$, and let $Z$ count members of $\mathcal F$ contained in $R$.
Proposition: The containment probability obeys \[ \Pr(Z>0)\ge\left(1+\frac{1-p}{pr}\right)^{-k} \ge 1-\frac{k(1-p)}{pr}. \]
Proof: The mean is $|\mathcal F|p^k$, and independence gives \[ \frac{\mathbb EZ^2}{(\mathbb EZ)^2} =\mathbb E_{A,B\in\mathcal F}p^{-|A\cap B|}. \] Fix $A$. Expand the product over its elements: \[ \mathbb E_B p^{-|A\cap B|} =\sum_{S\subseteq A}(p^{-1}-1)^{|S|}\Pr(S\subseteq B) \le\sum_{S\subseteq A}\left(\frac{1-p}{pr}\right)^{|S|} =\left(1+\frac{1-p}{pr}\right)^k. \] All terms are nonnegative. Cauchy--Schwarz applied to $Z\mathbf1_{Z>0}$ gives $(\mathbb EZ)^2\le\mathbb EZ^2\Pr(Z>0)$, proving the first bound. For $u\ge0$, the derivative of $(1+u)^{-k}-(1-ku)$ is $k[1-(1+u)^{-k-1}]\ge0$, proving the second. ∎
For $0<\varepsilon<1$, the elementary sufficient condition $pr\ge k/\varepsilon$ therefore gives failure probability at most $\varepsilon$. This is a weak spread-to-containment bound, not an improvement of modern sunflower thresholds. For example, all two-element subsets of a six-element universe satisfy the spread condition with $r=3$. At $p=1/2$, the first bound is $9/16$, while the exact containment probability is $57/64$. The attempted route toward sharper applications stops at the product bound: it uses only the separate upper bounds for each intersection event, discarding dependencies that stronger sunflower arguments exploit. Moreover, obtaining a suitably spread subfamily or core from an arbitrary family has not been proved here. The displayed estimate alone does not supply that extraction step.

Verification and remaining gaps

The six-element example was exhaustively checked over all subfamilies defined by containment and all 64 random subsets, verifying spread, both moments and the exact probability. The source's robust-sunflower, robust-daisy and spread variants are not declared equivalent. No quantitative improvement for clique circuits, random resolution or depth-three circuits is obtained.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2024/15. Relevant passages read: 2024/15 pp.898--899, sunflower variants and listed circuit/proof-complexity applications.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

Integration review.
Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, read and reviewed its written mathematical arguments, and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch025/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.

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