Abstract
In 1971 Trotter (or Bogart and Trotter) conjectured that every finite poset on at least 3 points has a pair whose removal does not decrease the dimension by more than 1. In 1992 Brightwell and Scheinerman introduced fractional dimension of posets, and they made a similar conjecture for fractional dimension. This paper settles this latter conjecture.
Highlights
1.1 Dimension of posetsLet P be a poset
The minimum cardinality of a realizer is called the dimension of the poset P, denoted by dim(P )
The following conjecture has become known as the Removable Pair Conjecture
Summary
The minimum cardinality of a realizer is called the dimension of the poset P , denoted by dim(P ). Every poset on at least 3 point has a removable pair. The origins of the conjecture are not entirely clear It appeared in print in a 1975 paper by Trotter [9], but according to Trotter [8], it was probably formulated at the 1971 summer conference on combinatorics held at Bowdoin College, and should be credited either to Trotter or to Bogart and Trotter. A linear extension L reverses the critical pair (x, y) if y < x in L. The following proposition expresses that the critical pairs are the only significant incomparable pairs for constructing realizers. A set of linear extensions is a realizer if and only if it reverses every critical pair
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