ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 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 pairing construction gives an Eventown family of size $2^{⌊ n / 2 ⌋}$

Example

For $n = 4$ , group the points into the pairs ${0, 1}$ and ${2, 3}$ . The family

$\emptyset, {0, 1}, {2, 3}, {0, 1, 2, 3}$

is Eventown. For $n = 5$ the same four sets, viewed as subsets of $[5]$ , are still Eventown and still maximal.

Facts & Assumptions

Given: the two families above.

[L1]

Eventown families have at most $2^{⌊ n / 2 ⌋}$ members (Eventown: distinct $A_{1}, \dots, A_{m} \subseteq [n]$ with every $∣ A_{i} ∣$ and every $∣ A_{i} \cap A_{j} ∣$ even satisfy $m \leq 2^{⌊ n / 2 ⌋}$ ).

[L2]

Maximal Eventown families have exactly that many members (An Eventown family that no further set can be added to has exactly $2^{⌊ n / 2 ⌋}$ members).

Verification

technique · direct

1.1given

In both displayed families every set has even size, and the intersection of any two displayed sets is again $\emptyset$ , ${0, 1}$ , ${2, 3}$ or ${0, 1, 2, 3}$ , hence even.

2.1L1step 1.1

For $n = 4$ there are four sets, which is $2^{2} = 2^{⌊ 4 / 2 ⌋}$ ; for $n = 5$ there are again four sets, which is $2^{2} = 2^{⌊ 5 / 2 ⌋}$ .

3.1L2step 2.1∎

The $n = 5$ family is maximal by [L2], so the floor in the general bound is visible already in the first odd case.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

20 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, §2.3.2 (standard reference, not scraped)

The pairing construction gives an Eventown family of size 2⌊n/2⌋2⌊n/2⌋