A sperner-type theorem

Let P=P ₁×P ₂×...×P _M be the direct product of symmetric chain orders P ₁, P ₂, ..., P _M . Let F be a subset of P containing no l+1 elements which are identical in M–1 components and linearly ordered in the Mth one. Then max |F|cM ^1/2lW(P), where W(P) is the cardinality of the largest level of P, and c is independent of P, M and l. Infinitely many P show that this result is best possible for every M and l apart from the constant factor c.     

Random graph orders

Let P _n be the order determined by taking a random graph G on {1, 2,..., n}, directing the edges from the lesser vertex to the greater (as integers), and then taking the transitive closure of this relation. We call such and ordered set a random graph order. We show that there exist constants c, and °, such that the expected height and set up number of P _n are sharply concentrated around cn and °n respectively. We obtain the estimates: .565<c<.610, and .034<°<.289. We also discuss the width, dimension, and first-order properties of P _n.     

Regressions and monotone chains: A ramsey-type extremal problem for partial orders

Aregression is a functiong from a partially ordered set to itself such thatg(x)≦x for allz. Amonotone k-chain is a chain ofk elementsx ₁<x ₂ <...<x _k such thatg(x ₁)≦g(x ₂)≦...≦g(x _k ). If a partial order has sufficiently many elements compared to the size of its largest antichain, every regression on it will have a monotone (k + 1)-chain. Fixingw, letf(w, k) be the smallest number such that every regression on every partial order with size leastf(w, k) but no antichain larger thanw has a monotone (k + 1)-chain. We show thatf(w, k)=(w+1) ^k . Dedicated to Paul Erdős on his seventieth birthday Research supported in part by the National Science Foundation under ISP-80-11451.     

Extremal problems for linear orders on [n] p

Let be a product order on [n] ^p i.e. for A, B [n] ^p , 1a ₁<a ₂<...<a _p º-n and 1<-b ₁<b ₂<...<b _p <-n we have AB iff a _i<-b _i for all i=1, 2,..., p. For a linear extension < of (ordering [n] ^p as ) let F _< ^[n],p (m) count the number of A _i 's, i<-m such that 1A _i. Clearly, for every m and <, where <l denotes the lexicographic order on [n] ^p . In this note we prove that the colexicographical order, <c, provides a corresponding lower bound i.e. that holds for any linear extension < of .This project together with [2] was initiated by the first author and continued in colaboration with the second author. After the death of the first author the work was continued and finalized by the second and the third author.Research supported by NSF grant DMS 9011850.     

Planar graphs and poset dimension

We view the incidence relation of a graph G=(V. E) as an order relation on its vertices and edges, i.e. a<_G b if and only of a is a vertex and b is an edge incident on a. This leads to the definition of the order-dimension of G as the minimum number of total orders on V E whose intersection is <_G. Our main result is the characterization of planar graphs as the graphs whose order-dimension does not exceed three. Strong versions of several known properties of planar graphs are implied by this characterization. These properties include: each planar graph has arboricity at most three and each planar graph has a plane embedding whose edges are straight line segments. A nice feature of this embedding is that the coordinates of the vertices have a purely combinatorial meaning.     

A theory of recursive dimension for ordered sets

The classical theorem of R. P. Dilworth asserts that a partially ordered set of width n can be partitioned into n chains. Dilworth's theorem plays a central role in the dimension theory of partially ordered sets since chain partitions can be used to provide embeddings of partially ordered sets in the Cartesian product of chains. In particular, the dimension of a partially-ordered set never exceeds its width. In this paper, we consider analogous problems in the setting of recursive combinatorics where it is required that the partially ordered set and any associated partition or embedding be described by recursive functions. We establish several theorems providing upper bounds on the recursive dimension of a partially ordered set in terms of its width. The proofs are highly combinatorial in nature and involve a detailed analysis of a 2-person game in which one person builds a partially ordered set one point at a time and the other builds the partition or embedding.This paper was prepared while the authors were supported, in part, by NSF grant ISP-80-11451. In addition, the second author received support under NSF grant MCS-80-01778 and the third author received support under NSF grant MCS-82-02172.     

A structural characterization of non-Arguesian lattices

This is the first of a planned series of papers on the structure of non-Arguesian modular lattices. Apart from the (subspace lattices of) non-Arguesian projective planes, the best known examples of such lattices are obtained via the Hall-Dilworth construction by badly gluing together two projective planes of the same order. Our principal result shows that every non-Arguesian modular lattice L retains some of the flavor of these examples: There exist in the ideal lattice of L 20 intervals, not necessarily distinct, that form non-degenerate projective plains, and 10 points and 10 lines in these planes that constitute in a natural sense a classical non-Arguesian configuration.Research supported by NSERC Operating Grant A8190.Research supported by NSF Grant DMS-8300107.     

Shellability of exponential structures

We show that the poset of all partitions of an nd-set with block size divisible by d is shellable. Using similar techniques, it also follows that various other examples of exponential structures cited by Stanley are also shellable. The method used involves the notion of recursive atom orderings introduced by Björner and Wachs.Research supported in part by NATO post-doctoral grant administered by the NSF.     

On the greedy dimension of a partial order

This paper introduces a new concept of dimension for partially ordered sets. Dushnik and Miller in 1941 introduced the concept of dimension of a partial order P, as the minimum cardinality of a realizer, (i.e., a set of linear extensions of P whose intersection is P). Every poset has a greedy realizer (i.e., a realizer consisting of greedy linear extensions). We begin the study of the notion of greedy dimension of a poset and its relationship with the usual dimension by proving that equality holds for a wide class of posets including N-free posets, two-dimensional posets and distributive lattices.     

Substitution and atomic extension on greedy posets

In this paper, we study the operations of substitution and atomic extension on greedy posets. For the substitution operation, if P=(P ₁ , x, P ₂ )is a greedy poset, then P ₁ and P ₂ are greedy posets, the converse being false. However, for the atomic extension, P=P ₁ (x, P ₂ )is a greedy poset if and only if P ₁ and P ₂ are greedy posets. We prove also that the class of greedy semi-partitive lattices is the smallest one containing M _n (n2), B ₃ and closed by atomic extension. The class C _n of greedy posets with jump number n is infinite. However, we show that C _n can be obtained, in a very simple way, from a subclass D _n of finite cardinal ity. We construct D _n for n=1, 2.     

Characteristic for reflexive relations

Poincaré characteristic for reflexive relations (oriented graphs) is defined in terms of homology and is not invariant under transitive closure. Formulas for the Poincaré characteristic of products, joins, and bounded products are given. Euler's definition of characteristic extends to certain filtrations of reflexive relations which exist iff there are no oriented loops. Euler characteristic is independent of filtration, agrees with Poincaré characteristic, and is unique. One-sided Möbius characteristic is defined as the sum of all values of a one-sided inverse of the zeta function. Such one-sided inverses exist iff there are no local oriented loops (although there may be global oriented loops). One-sided Möbius characteristic need not be Poincaré characteristic, but it is when a one-sided local transitivity condition is satisfied. A two-sided local transitivity condition insures the existence of the Möbius function but Möbius inversion fails for non-posets.     

Angle orders

A finite poset is an angle order if its points can be mapped into angular regions in the plane so thatx precedesy in the poset precisely when the region forx is properly included in the region fory. We show that all posets of dimension four or less are angle orders, all interval orders are angle orders, and that some angle orders must have an angular region less than 180° (or more than 180°). The latter result is used to prove that there are posets that are not angle orders.The smallest verified poset that is not an angle order has 198 points. We suspect that the minimum is around 30 points. Other open problems are noted, including whether there are dimension-5 posets that are not angle orders.Research supported in part by the National Science Foundation, grant number DMS-8401281.     

Angle orders,regular n-gon orders and the crossing number

A finite poset P(X,<) on a set X={ x ₁,...,x _m} is an angle order (regular n-gon order) if the elements of P(X,<) can be mapped into a family of angular regions on the plane (a family of regular polygons with n sides and having parallel sides) such that x _ij if and only if the angular region (regular n-gon) for x _i is contained in the region (regular n-gon) for x _j. In this paper we prove that there are partial orders of dimension 6 with 64 elements which are not angle orders. The smallest partial order previously known not to be an angle order has 198 elements and has dimension 7. We also prove that partial orders of dimension 3 are representable using equilateral triangles with the same orientation. This results does not generalizes to higher dimensions. We will prove that there is a partial order of dimension 4 with 14 elements which is not a regular n-gon order regardless of the value of n. Finally, we prove that partial orders of dimension 3 are regular n-gon orders for n3.This research was supported by the Natural Sciences and Engineering Research Council of Canada, grant numbers A0977 and A2415.     

Circle orders,N-gon orders and the crossing number

Let ={P ₁,...,P _m } be a family of sets. A partial order P(, <) on is naturally defined by the condition P _i <P _j iff P _i is contained in P _j . When the elements of are disks (i.e. circles together with their interiors), P(, <) is called a circle order; if the elements of are n-polygons, P(, <) is called an n-gon order. In this paper we study circle orders and n-gon orders. The crossing number of a partial order introduced in [5] is studied here. We show that for every n, there are partial orders with crossing number n. We prove next that the crossing number of circle orders is at most 2 and that the crossing number of n-gon orders is at most 2n. We then produce for every n4 partial orders of dimension n which are not circle orders. Also for every n>3, we prove that there are partial orders of dimension 2n+2 which are not n-gon orders. Finally, we prove that every partial order of dimension 2n is an n-gon order.This research was supported under Natural Sciences and Engineering Research Council of Canada (NSERC Canada) grant numbers A2507 and A0977.     

Two combinatorial problems on posets

Bialostocki proposed the following problem: Let nk2 be integers such that k|n. Let p(n, k) denote the least positive integer having the property that for every poset P, |P|p(n, k) and every Z _k -coloring f: P Z _k there exists either a chain or an antichain A, |A|=n and _aA f(a) 0 (modk). Estimate p(n, k). We prove that there exists a constant c(k), depends only on k, such that (n+k–2)²–c(k) p(n, k) (n+k–2)²+1. Another problem considered here is a 2-dimensional form of the monotone sequence theorem of Erdös and Szekeres. We prove that there exists a least positive integer f(n) such that every integral square matrix A of order f(n) contains a square submatrix B of order n, with all rows monotone sequences in the same direction and all columns monotone sequences in the same direction (direction means increasing or decreasing).     

A cofinal coloring theorem for partially ordered algebras

If P is a directed partially ordered algebra of an appropriate sort-e.g. an upper semilattice-and has no maximal element, then P has two disjoint subalgebras each cofinal in P. In fact, if P has cofinality then there exists a family of such disjoint subalgebras. A version of this result is also proved without the directedness assumption, in which the cofinality of P is replaced by an invariant which we call its global cofinality.This work was done while the first author was partly supported by NSF contract MCS 82-02632.     

The uniqueness of the core

Simple proofs are given for two theorems of Duffus and Rival: If a finite poset is dismantled by irreducibles as much as possible, the subposet one finally obtains is unique up to isomorphism. If one dismantles by doubly irreducibles, the subposet is unique.     

Decompositions of partially ordered sets into chains and antichains of given size

Every partially ordered set P on at least (1+o(1))n ³ elements can be decomposed into subposets of size n that are almost chains or antichains. This lower bound on P is asymptotically best possible. Similar results are presented for other types of combinatorial structures.Research supported in part by the AKA Research Fund of the Hungarian Academy of Sciences, grant 1-3-86-264.     

Cross-monotone subsequences

Two finite real sequences (a ₁,...,a _k ) and (b ₁,...,b _k ) are cross-monotone if each is nondecreasing anda _i+1–a _i b _i+1–b _i for alli. A sequence (₁,..., _n ) of nondecreasing reals is in class CM(k) if it has disjointk-term subsequences that are cross-monotone. The paper shows thatf(k), the smallestn such that every nondecreasing (₁,..., _n ) is in CM(k), is bounded between aboutk ²/4 andk ₂/2. It also shows thatg(k), the smallestn for which all (₁,..., _n ) are in CM(k)and eithera _k b ₁ orb _k a ₁, equalsk(k–1)+2, and thath(k), the smallestn for which all (₁,..., _n ) are in CM(_k)and eithera ₁b ₁...a _k b _k orb ₁a ₁...b _k a _k , equals 2(k–1)²+2.The results forf andg rely on new theorems for regular patterns in (0, 1)-matrices that are of interest in their own right. An example is: Every upper-triangulark ²×k ² (0, 1)-matrix has eitherk 1's in consecutive columns, each below its predecessor, ork 0's in consecutive rows, each to the right of its predecessor, and the same conclusion is false whenk ² is replaced byk ²–1.     

Constructing greedy linear extensions by interchanging chains

A natural way to prove that a particular linear extension of an ordered set is ‘optimal’ with respect to the ‘jump number’ is to transform this linear extension ‘canonically’ into one that is ‘optimal’. We treat a ‘greedy chain interchange’ transformation which has applications to ordered sets for which each ‘greedy’ linear extension is ‘optimal’.