We will talk here about POSETS – partially ordered sets. The study of order is very important in many areas of mathematics starting with the order relation on the integers and reals in algebra and in Euclidean geometry. The set of all subsets of a set can be partially ordered by inclusion and this is a very basic example of posets. While the study of order and posets is a separate area on its own, parts of it are very important in extremal combinatorics and we will give a little taste here.
Dear readers, please contribute your favorite result or problem on partially ordered sets (or Sorting) in the comment session.
A chain in a POSET is a set of elements so that every two of them are comparable. An antichain $A \subset P$ is a set of elements so that every two distinct elemenאs in are incomparable. (Antichains are also called independent sets.) An immediate but important Lemma is:
The immediate lemma: The intersection of a chain and an antichain contains at most one element. Walla!
Dilworth’s theorem (DT): Every finite partially ordered set can be covered by chains.
(By the immediate lemma, at least chains are needed.)
Dual Dilworth theorem: Every partially ordered sets can be covered by antichains.
(By the immediate lemma, at least antichains are needed.)
The proof of the dual Dilworth theorem is easy. Note that the set of minimal elements of is an antichain. Let . We need two easy observations. First, and second: If then there is a chain with one element from . Walla!
The proof of Dilworth’s theorem is by induction on $|P|$. For the induction step you first consider the case where every antichain of maximal size is either or . In this case you consider a chain with one element in and one element in and delete these elements from . For the resulting post , and we can use the induction hypothesis.
Otherwise there is an antichain of maximum size which is not MAX(P) or MIN(P). Put . Let be the set of elements in which are larger or equal some element in , and let be the set of elements in which are smaller or equal some element in .
- . Otherwise we could add an element to to form a larger antichain.
- . Otherwise, there will be two elements of which are comparable.
So by the induction hypothesis can be covered by chains and can be covered by $ chains . Bu re-indexing we can assume that both and contains . It follows that is the minimal element in $C_i^+$ and the maximal element in $C_i^-$ and hence is a chain. The chains cover . Sababa!
An important Corollary both from Dilworth’s theorem and its dual is that
The fundamental Erdos Szekeres theorem asserts that if then every sequence of different real numbers contains a monotone increasing sequence of length or a monotone decreasing sequence of length .
There are simple proofs. For example, associate to every a pair of integers where is the maximum length of the increasing subsequence starting with and is the maximum length of the deccreasing subsequence starting with . The result follows from the easy observation that all these pairs are different.
Both Dilworth’ theorem and its easy dual imply easily (in fact we need only the important corollary) the Erdos Szekeres theorem when we define the following partial order: if both and .
Looking at Sperner’s theorem again
Sperner’s theorem asserts that the maximal size of an antichain of subsets of an elements set is . By Dilworth’s theorem it follows that we can cover all sets by chains (and, of course when we exhibit such a covering it reproves Sperner’s theorem). A symmetric saturated chain decomposition is a partition of (=all subsets of ) to saturated chains where each chain has, for some , sets of sizes $k,k+1,\dots,d-k$. You can build such a decomposition inductively.
Start with a decomposition for for each chain create a new chain by adding the element to every set. And then move the top set in to . Walla!
This is the beginning of a very beautiful story related also to the Dedekind Problem about number of antichains in .
Curtis Greene and Danny Kleitman
The Greene-Kleitman theorem
Let be the maximum size of the union of antichains in a poset . For every chain For every chain we have . Therefore for a partition of to chains we have . The Greene-Kleitman theorem asserts that there is always a decomposition into chains with .
The perfect graph theorem.
What is the relation between the very easy dual Dilworth theorem and the harder Dilworth theorem? As it turns out there is a very general theorem, Lovasz’ perfect graph theorem, that shows that these two theorems are equivalent.
A graph G is perfect if for every induced subgraph H, the chromatic number equals the clique number. Lovasz’ theorem (conjectured by Claude Berge) asserts that complements of perfect graphs are perfect. The perfectness of the comparability graph of a poset amounts to the dual Dilworth theorem, and for its complement it is the Dilworth theorem. Lovasz in fact proved that perfectness is equivalent to the relation $\latex \omega(H)\cdot \alpha (H) \ge |H|$ for every induced subgraph H. (For posets this is our important corollary above.)
Jeff Kahn and Jake Baron (see here on their 2016 asymptotic solution to Tusza’s conjecture), Mike Saks, and Nati Linial
Startling theorems on POSETS: Kahn-Saks, Linial-Saks, Linial-Kahn, and Kahn-Saks
Here are some beautiful and important theorems on posets. An order ideas of a post is a set of elements so that if and then .
Theorem (Linial-Saks, 1985): In every poset there is an element which is contained in more than δ and less than 1-δ order ideas of . (Paper)
Theorem (Kahn-Saks, 1984): For every Poset which is not a chain there are two incomparable elements such that the number of linear extensions of for which is between 3/11 and 8/11. (Paper)
A simpler proof was found in the late 80s by Kahn and Linial and by Karzanov and Khachiyan. It is based on the Brunn Minkowski theorem and gives a weaker constant .
Theorem (Kahn-Saks, 1987): For every finite distributive lattice the maximum antichain is of size . (Paper)
Lattices are special types of posets with the property that for every set of elements (pairs suffice in the finite case), there is a unique minimal elements above them all (denoted for pairs by x ∧ y) and a unique maximal element (denoted for pairs by x ∨ y) below them all.
A distributive lattice is a lattice that satisfies for every x, y and z, the relation
x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z)
Birkhoff’s representation theorem asserts that finite distributive lattices can be represented as order ideals of posets (ordered by inclusion).