CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + 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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The $n$ singletons form an Oddtown family, so the bound $m \leq n$ is attained for every $n$

Statement

For every natural number $n$ , the singleton family

$({0}, {1}, \dots, {n - 1})$

is an Oddtown family on $[n]$ . Hence the bound $m \leq n$ of Oddtown: distinct $A_{1}, \dots, A_{m} \subseteq [n]$ with every $∣ A_{i} ∣$ odd and every $∣ A_{i} \cap A_{j} ∣$ ( $i \neq j$ ) even satisfy $m \leq n$ is sharp.

Facts & Assumptions

Given: a natural number $n$ .

[L1]

Distinct subsets of $[n]$ whose sizes are odd and whose pairwise intersections have even size number at most $n$ (Oddtown: distinct $A_{1}, \dots, A_{m} \subseteq [n]$ with every $∣ A_{i} ∣$ odd and every $∣ A_{i} \cap A_{j} ∣$ ( $i \neq j$ ) even satisfy $m \leq n$ ).

Proof

technique · direct

1.1

Each singleton has odd size $1$ , and any two distinct singletons meet in the empty set, whose size is even. So the family satisfies the hypotheses of [L1].

L1given

2.1

The family has exactly $n$ members, so it attains the bound of [L1].

step 1.1∎

Remarks

This is the standard extremal example, and the companion page uses it to refute the false improvement $m \leq n - 1$ .

Depends on

Used by

An Oddtown family of four clubs on four citizens, and why a fifth cannot be added Example
FALSE: an Oddtown family on [n] has at most n-1 members False statement

Dependency tree · two levels

19 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, §1.1 (standard reference, not scraped)

The nn singletons form an Oddtown family, so the bound m≤nm≤n is attained for every nn