Introduction to Posets

By Tom Trotter
Arizona State University

Alternate Definition

Definition. Alternatively, dim(P) is the least t so that
P is isomorphic to a:

Is G planar ?

(W. Schnyder'89)


G is planar

dim P(G) <= 3





The Incidence Poset of a Graph

Definition. Given a graph G = (V, E), we associate a poset whose ground set is V U E. The order is defined by x < S if and only if xE V, SEE andxES.

A Probability Space

The probability, that x < y in P, denoted Pr[x < y], is the number of linear extensions with x < y over the total number of linear



Pr[x < y] = 2/5.

The XYZ Theorem

(Sh.epp '80)

Pr[x > y]<= Pr[x > ylx > z ]

The Strict XYZ Theorem

(Fishburn '84)

If x and z form an antichain,


Pr[x > y] < Pr[x > ylx > z ]

The 1/3 - 2/3 Conjecture

(Kislitsyn '66)

If P is a poset which

is no a chain, then there exists X,Y in P so that

1/3 < Pr[x < y] < 2/3

Best possible

If true!

Pr[a < b] = Pr[b < c]= 2/3

Theorem (Kahn, Saks '84)

If P is a poset which

is not a chain, then there

exists X, y in P so that
3/11 < Pr X < y < 8/11

Note.* Proof requires the

Alexandrov/Fenchel inequalities for mixed volumes,.

Theorem (Brightwell, FeIsner and, Trotter '95)

If P is a poset which .

is not a chain, then there exists x, y in P so that

Note: Proof builds on Kahn and Saks approach. Also requires Ahlswede/Daykin Four Function Theorem.