Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Finite cutsets and finite antichains

  • 43 Accesses

  • 21 Citations


An ordered set (P,≤) has the m cutset property if for each x there is a set Fx with cardinality less than m, such that each element of Fx is incomparable to x and {x} ∪ Fx meets every maximal chain of (P,≤). Let n be least, such that each element x of any P having the m cutset property belongs to some maximal antichain of cardinality less than n. We specify n for m < w. Indeed, n-1=m= width P for m=1,2,n=5 if m=3 and n⩾ℵ1 if m ≥4. With the added hypothesis that every bounded chain has a supremum and infimum in P, it is shown that for 4⩽m⩽ℵ0, n=ℵ0. That is, if each element x has a finite cutset Fx, each element belongs to a finite maximal antichain.

This is a preview of subscription content, log in to check access.


  1. 1.

    J. Bell and J. Ginsburg (to appear) Compact spaces and spaces of maximal complete subgraphs, Transactions AMS.

  2. 2.

    R. P.Dilworth (1950) A decomposition theorem for partially ordered sets, Ann. Math. 51, 161–166.

  3. 3.

    J. Ginsburg (to appear) Compactness and subsets of ordered sets that meet every maximal chain.

  4. 4.

    J. Ginsburg, I. Rival, and E. W. Sands (to appear) Antichains and finite sets that meet all maximal chains.

Download references

Author information

Additional information

This work was supported by the NSERC of Canada.

Communicated by I. Rival

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sauer, N., Woodrow, R.E. Finite cutsets and finite antichains. Order 1, 35–46 (1984). https://doi.org/10.1007/BF00396272

Download citation

AMS (MOS) subject classifications (1980)

  • 06A10

Key words

  • Ordered set
  • chain
  • antichains
  • width
  • cutset