False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-07-31

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Literature-sourced — the exact statement appears in a cited source; only wording and notation differ.
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False: every maximal antichain in a finite poset has maximum cardinality

Statement

Every maximal antichain in a finite poset has cardinality equal to the width.

Facts & Assumptions

Given: The three-element poset $P=\{a,b,c\}$ with $a<b$ , $a<c$ , and $b,c$ incomparable.

[F1]

A maximal antichain is inclusion-maximal, while a maximum antichain has greatest cardinality; the width is the cardinality of a maximum antichain (Antichains, chain covers, and antichain covers of a poset, Height and width of a nonempty finite poset).

Refutation

technique · direct

1.1

The singleton $\{a\}$ is an antichain, and it is maximal because both remaining elements $b$ and $c$ are comparable with $a$ .

givenF1

1.2

The set $\{b,c\}$ is an antichain of cardinality $2$ , so the width of $P$ is at least $2$ and $\{a\}$ is not maximum.

givenF1

2.1

Thus the finite poset $P$ has a maximal antichain that is not maximum, refuting the Statement.

step 1.1step 1.2∎

Depends on

Used by

A maximal antichain of size one in a finite poset of width two Counterexample

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

M. Keller and W. T. Trotter, Applied Combinatorics, §6.4 (standard reference, not scraped)