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

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

Statement refuted

The false statement False: every maximal antichain in a finite poset has maximum cardinality claims that every maximal antichain in a finite poset has maximum cardinality.

Facts & Assumptions

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

[F1]

An antichain is maximal when no larger antichain contains it, and maximum when no antichain has greater cardinality; the width is the maximum cardinality of an antichain (Antichains, chain covers, and antichain covers of a poset, Height and width of a nonempty finite poset).

Counterexample

technique · direct

1.1

The singleton $\{a\}$ is an antichain and cannot be enlarged, since both $b$ and $c$ are comparable with $a$ .

givenF1

1.2

The pair $\{b,c\}$ is an antichain. No three-element antichain exists, because the only three-element subset is $P$ itself and it contains the comparable pair $a,b$ ; hence the width of $P$ is $2$ .

givenF1

2.1

Thus $\{a\}$ is maximal of cardinality $1$ but not maximum, providing the required counterexample.

step 1.1step 1.2∎

Remarks

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 24 results over 12 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)