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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A $k$ -uniform family on $[n]$ with all pairwise intersections of size $t < k$ has at most $n$ members

Statement

Let $A_{1}, \dots, A_{m}$ be distinct subsets of $[n]$ , each of size $k$ , and suppose

$∣ A_{i} \cap A_{j} ∣ = t < k for every i \neq j .$

Then $m \leq n$ .

Facts & Assumptions

Given: a $k$ -uniform family $A_{1}, \dots, A_{m} \subseteq [n]$ with every pairwise intersection of size $t < k$ .

[L1]

The nonuniform Fisher inequality gives $m \leq n$ for distinct nonempty sets with constant pairwise intersection size (Fisher's inequality, nonuniform form: distinct nonempty $A_{1}, \dots, A_{m} \subseteq [n]$ with $∣ A_{i} \cap A_{j} ∣ = t$ for all $i \neq j$ satisfy $m \leq n$ ).

Proof

technique · direct

1.1

Every set in the family has size $k > t$ , so the hypotheses place the family in the second case of [L1].

L1given

2.1

Applying [L1] to that case gives $m \leq n$ .

L1step 1.1∎

Remarks

This is the design-theoretic reading of Fisher's inequality, stated without importing any block-design terminology onto the page.

Depends on

Used by

The seven lines of the Fano plane meet pairwise in one point, and Fisher's bound is tight Example

Dependency tree · two levels

18 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)

A kk-uniform family on [n][n] with all pairwise intersections of size t<kt<k has at most nn members