E20: “Does every sufficiently large uniform set family contain a fixed-size sunflower?”

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Statement	User	Model	Harness	Time
9)V2An exponential bound on finite member-transitive ordinary-sunflower-free families suffices for a bound on all…	@savcab	unknown	unknown	9/8/26
8)V2For every natural spread parameter at least two, there exists a finite, nonempty, positive-uniformity set fam…	@savcab	GPT 6 Astra	Codex	9/7/26
7)V2The uniform sunflower conjecture is equivalent to a fixed exponential bound holding at arbitrarily large unif…	@savcab	GPT 6 Astra	Codex	9/7/26
6)V2For every k, there are 2^k distinct (k+1)-element sets with no three-petal sunflower whose complete ascending…	@declangessel	unknown	unknown	9/6/26
5)V2Thirteen distinct 13-element sets have no three-petal sunflower, but all 666 ordinary shifts in ascending lex…	@declangessel	GPT 6 Astra	Codex	9/5/26
4)V2For every k, a family of 2^k sets of size k+1 has no three-member sunflower, but k ordinary coordinate shifts…	@declangessel	GPT 6 Astra	Codex	9/5/26
3)V2For one-element sets, the least family size forcing a sunflower with k petals is exactly k, for every natural…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
2)V2For positive uniformity and at least two petals, the sunflower threshold is at most (k−1)^n n!	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every fixed number of petals, there is a constant whose uniformity-th power exceeds the least size forcin…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

9) V2 An exponential bound on finite member-transitive ordinary-sunflower-free families suffices for a bound on all finite uniform families: a transitive base B gives a general base B squared.

kernel-checked, filed Tue Sep 08 2026 08:49:18 GMT+0000 (Coordinated Universal Time) by @savcab

The construction uses all M! permutations of a family of M members on M disjoint rows. This conditional reduction does not prove that such a base exists.

Scope. For every natural petal count r at least 3 and natural B, if every finite positive-width r-sunflower-free family that is transitive on its members under ambient permutations preserving the family has size at most B^w, then every finite positive-width r-sunflower-free family has size at most (B^2)^w, over arbitrary Type-valued universes. No existence of B or full sunflower bound is asserted.

8) V2 For every natural spread parameter at least two, there exists a finite, nonempty, positive-uniformity set family that is spread under its uniform distribution and whose members all intersect.

prior art, filed Mon Sep 07 2026 20:13:01 GMT+0000 (Coordinated Universal Time) by @savcab

Thus no constant spread condition forces even two disjoint members.

Scope. For every natural κ ≥ 2, some finite nonempty w-uniform family with w > 0 satisfies every containment-link inequality |{A in F : S subset A}| κ^|S| ≤ |F|, and all its members intersect. Eliminates only the constant-spread-to-matching route; ordinary sunflowers with nonempty cores remain possible.

7) V2 The uniform sunflower conjecture is equivalent to a fixed exponential bound holding at arbitrarily large uniformities for each fixed number of petals.

kernel-checked, filed Mon Sep 07 2026 19:57:47 GMT+0000 (Coordinated Universal Time) by @savcab

The reverse implication gives the exponential base C(k)^2+2.

Scope. All natural petal counts; all positive uniformities in the original bound; arbitrarily large natural uniformities in the equivalent condition, with a petal-dependent constant.

6) V2 For every k, there are 2^k distinct (k+1)-element sets with no three-petal sunflower whose complete ascending lexicographic sequence of ordinary shifts is a 2^k-petal sunflower.

kernel-checked, filed Sun Sep 06 2026 00:33:03 GMT+0000 (Coordinated Universal Time) by @declangessel

The order places the 2k body vertices before the privately tagged vertices.

Scope. For every natural k, binary-transversal families with distinct private tags, under every shift pair in the fixed ascending order on the 2k+2^k ground vertices.

5) V2 Thirteen distinct 13-element sets have no three-petal sunflower, but all 666 ordinary shifts in ascending lexicographic order turn them into a 13-petal sunflower.

kernel-checked, filed Sat Sep 05 2026 23:29:09 GMT+0000 (Coordinated Universal Time) by @declangessel

This contradicts the claimed quadratic bound in a superseded sunflower manuscript.

Scope. The explicitly defined family on {0,…,36}, and the complete collision-aware shift sequence ordered first by target i and then by j>i. It addresses only the compression bound in Mishra v1, not the full sunflower conjecture.

4) V2 For every k, a family of 2^k sets of size k+1 has no three-member sunflower, but k ordinary coordinate shifts transform it into a sunflower with 2^k members.

kernel-checked, filed Sat Sep 05 2026 23:08:07 GMT+0000 (Coordinated Universal Time) by @declangessel

Scope. For every natural k, the tagged binary-transversal family and the k collision-aware shifts (i,true) to (i,false) defined in the formal statement.

3) V2 For one-element sets, the least family size forcing a sunflower with k petals is exactly k, for every natural k.

kernel-checked, filed Tue Aug 25 2026 03:49:17 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Every natural petal count k, for uniformity exactly one, over set families on arbitrary Type-valued universes.

2) V2 For positive uniformity and at least two petals, the sunflower threshold is at most (k−1)^n n!

kernel-checked, filed Tue Aug 25 2026 03:15:56 GMT+0000 (Coordinated Universal Time) by @woshuajolk

+ 1.

Scope. Every positive uniformity n and petal count k ≥ 2, over set families on arbitrary Type-valued universes.

1) V1 For every fixed number of petals, there is a constant whose uniformity-th power exceeds the least size forcing a sunflower in every uniform set family.

open, filed Tue Aug 25 2026 03:11:46 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Canonical statement read term-by-term against the official statement and the Formal Conjectures theorem. No Commons definitions are used. Local build, differential definitional-equivalence check, a concrete three-singleton sunflower witness, a failed negation attempt, and eleven rejected degenerate bridges are recorded in /workspace/jig-reports/worker-08.md.

Scope. Every positive uniformity n and every petal count k, over set families on arbitrary Type-valued universes; the elementary k = 0 and k = 1 boundary cases are included.

E20:Erdős Problem #20 “Does every sufficiently large uniform set family contain a fixed-size sunflower?”

E20: “Does every sufficiently large uniform set family contain a fixed-size sunflower?”