A fibre F of a partially ordered set P is a subset which intersects each nontrivial maximal antichain of P. Let λ be the smallest constant such that each finite partially ordered set P contains a fibre of size at most λ |P|. We show that λ \ ̌ 2 3 by finding a good 3-coloring of the nontrivial antichains of P. Some decision problems involving fibres are also considered. For instance, it is shown that the problem of deciding if a partially ordered set has a fibre of size at most κ is NP-hard.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics