Posets with large dimension and relatively few critical pairs

The dimension of a poset (partially ordered set)P=(X, P) is the minimum number of linear extensions ofP whose intersection isP. It is also the minimum number of extensions ofP needed to reverse all critical pairs. Since any critical pair is reversed by some extension, the dimensiont never exceeds the number of critical pairsm. This paper analyzes the relationship betweent andm, when 3⩽t⩽m⩽t+2, in terms of induced subposet containment. Ifm⩽t+1 then the poset must containSt, the standard example of at-dimensional poset. The analysis form=t+2 leads to dimension products and David Kelly's concept of a split. Whent=3 andm=5, the poset must contain eitherS3, or the 6-point poset called a chevron, or the chevron's dual. Whent⩾4 andm=t+2, the poset must containSt, or the dimension product of the Kelly split of a chevron andSt−3, or the dual of this product.