CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck 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


\node[font=\scriptsize,anchor=north] at (0,-.65) {maximal antichain $\{a\}$}; \draw[blue!75!black,dashed] (-2.05,2.2)--(2.05,2.2); \node[font=\scriptsize,anchor=south] at (0,2.27) {maximum antichain $\{b,c\}$}; \end{tikzpicture} ```

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

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