Bill Sands

On monochromatic paths in edge-coloured digraphs☆

Abstract Let G be a directed graph whose edges are coloured with two colours. Call a set S of vertices of Gindependent if no two vertices of S are connected by a monochromatic directed path. We prove that if G contains no monochromatic infinite outward path, then there is an independent set S of vertices of G such that, for every vertex x not in S, there is a monochromatic directed path from x to a vertex of S. In the event that G is infinite, the proof uses Zorns lemma. The last part of the paper is concerned with the case when G is a tournament.

Two-colouring all two-element maximal antichains

Abstract A fibre in a partially ordered set P is a subset of P meeting every maximal antichain of P . We give an example of a finite poset P with no one-element maximal antichain and containing no fibre of size at most | P }2, thus answering a question of Aigner-Andreae and disproving a conjecture of Lonc-Rival. We also prove Theorem 1. The elements of an arbitrary partially ordered set can be coloured with two colours such that every two-element maximal antichain receives both colours .

Lexicographic matchings cannot form Hamiltonian cycles

For any positive integer k let B(k) denote the bipartite graph of k- and k+1-element subsets of a 2k+1-element set with adjacency given by containment. It has been conjectured that for all k, B(k) is Hamiltonian. Any Hamiltonian cycle would be the union of two (perfect) matchings. Here it is shown that for all k>1 no Hamiltonian cycle in B(k) is the union of two lexicographic matchings.

Enumeration of order preserving maps

Three results are obtained concerning the number of order preserving maps of an n-element partially ordered set to itself. We show that any such ordered set has at least 22n/3 order preserving maps (and 22 in the case of length one). Precise asymptotic estimates for the numbers of self-maps of crowns and fences are also obtained. In addition, lower bounds for many other infinite families are found and several precise problems are formulated.

Counting antichains in finite partially ordered sets

We prove that for each integer l > 1 there exists a number r = r(l) > 1 such that every finite poset P of length l - 1 contains an element a satisfying no. of antichains of P containing atotal no. of antichains of P>=1r For the case l = 2, r = 8.807 will do. A consequence is that every finite distributive lattice L whose poset of join-irreducibles has length one contains a prime ideal I satisfying 19 < |I|/|L| < 89. In the other direction, we show that r(2) cannot be chosen less than 4.3865297.

A note on the congruence lattice of a finitely generated algebra

The study of finitely generated algebras is important to the theory of algebraic systems. Problems concerning free algebras and varieties of algebras often turn on this theme. At the same time properties of the lattice of congruence relations of an algebra often bear immediate consequences for the structure theory of an algebraic system. Let 9t be an algebra, that is, a pair , where A is a set and F is a family of operations on A. We say that 9t is of finite type if I FI is finite. Let Con(W) denote the lattice of all congruence relations of 9t partially ordered by set inclusion. For elements x and y of A let 0 (x, y) denote the principal congruence relation generated by identifying x and y; that is, 9 (x, y) = 0OIO E Con(%) and x _ y(9)). For a general reference to terminology we refer the reader to [1] or [3].

An inequality for the sizes of prime filters of finite distributive lattices

Abstract Let L be a finite distributive lattice, and let J(L) denote the set of all join-irreducible elements of L. Set J(L) = |J(L)|. For each a ∈J(L), let u(a) denote the number of elements in the prime filter {x ∈L: x⩾a} Our main theorem is Theorem 1. For any finite distributive lattice L, ∑ a ∈J(L) 4 u(a) ⩾j(L)4 |L| 2 . The base 4 here can most likely be replaced by a smaller number, but it cannot be replaced by any number strictly between 1 and 1.6159. We also make a few other observations about prime filters and the numbers u(a), a ∈ J(L), among which is: every finite distributive non-Boolean lattice L contains a prime filter of size at most |L|/3 or at least 2|L|/3. The above inequality is certainly not true for all finite lattices. However, we give another inequality, equivalent to the above for distributive lattices, which might hold for all finite lattices. If so, this would give an immediate proof of a conjecture known as Frankls conjecture.

Lattices of lattice homomorphisms

IfL andM are lattices, Hom (L, M) denotes the set of homomorphisms ofL intoM with the pointwise partial order.L is calledcatalytic if Hom (L, M) is a lattice for every latticeM. Among other results, it is shown that every retract of a lattice completely freely generated by a partially ordered set is catalytic, and that every catalytic lattice is semidistributive and satisfies Whitmans condition (W).

Chromatic Numbers and Homomorphisms of Large Girth Hypergraphs

We consider the problem of determining the minimum chromatic number of graphs and hypergraphs of large girth which cannot be mapped under a homomorphism to a specified graph or hypergraph. More generally, we are interested in large girth hypergraphs that do not admit a vertex partition of specified size such that the subhypergraphs induced by the partition blocks have a homomorphism to a given hypergraph. In the process, a general probabilistic construction of large girth hypergraphs is obtained, and general definitions of chromatic number and homomorphisms are considered.

Minimum sized fibres in distributive lattices

A subset F of an ordered set X is a fibre of X if F intersects every maximal antichain of X . We find a lower bound on the function ƒ ( D ), the minimum fibre size in the distributive lattice D , in terms of the size of D . In particular, we prove that there is a constant c such that In the process we show that minimum fibre size is a monotone property for a certain class of distributive lattices. This fact depends upon being able to split every maximal antichain of this class of distributive lattices into two parts so that the lattice is the union of the upset of one part and the downset of the other.