CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The complementary $2$ - $(7, 4, 2)$ design is symmetric but not a projective plane

Statement refuted

Every symmetric $2$ -design is a projective plane.

Facts & Assumptions

Given: The seven Fano lines on $Z / 7$ from The seven translates of ${0, 1, 3}$ form the Fano plane.

[L1]

Every projective plane of order $n$ is a symmetric $2$ - $(n^{2} + n + 1, n + 1, 1)$ design (A projective plane of order $n$ is a symmetric $2$ - $(n^{2} + n + 1, n + 1, 1)$ design).

Counterexample

technique · direct

1.1

Let $C$ be the family of complements of the seven Fano lines inside the seven-point set $Z / 7$ . Each member of $C$ has size $4$ , and there are seven of them.

givenalgebra

1.2

Fix distinct points $a, b \in Z / 7$ . Among the seven displayed Fano lines, exactly one contains both $a$ and $b$ , and inspection of the list shows that each of $a$ and $b$ lies on exactly three lines. So exactly five lines meet ${a, b}$ , and therefore exactly two Fano lines avoid both points. Hence exactly two members of $C$ contain ${a, b}$ , so $C$ is a $2$ - $(7, 4, 2)$ design.

givenalgebra

2.1

Since $C$ has seven blocks on seven points, it is symmetric. If it were a projective plane, then [L1] would make it a symmetric $2$ - $(7, 4, 1)$ design, contradicting step 1.2. Therefore this symmetric design is not a projective plane.

step 1.2L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, §4.1 (standard reference, not scraped)

The complementary 22-(7,4,2)(7,4,2) design is symmetric but not a projective plane