E. C. Milner

Unfriendly partitions of a graph

Minnesota, Minneapoli.~, Minnesota Communicated by the Managing Editors Received March 8. 1988 It has been conjectured by Cowan and Emerson [3] that every graph has an unfriendly partition; i.e., there is a partition of the vertex set V= V, v V, such that every vertex of V, is joined to at least as many vertices in V, _, as to vertices in V,. It is easily seen that every rinite graph has such a partition, and hence by compact- ness so does any locally finite graph. We show that the conjecture is also true for graphs which satisfy one of the following two conditions: (i) there are only finitely many vertices having infinite degrees; (ii) there are a finite number of infinite cardinals “to < ntI < cm, such that m, is regular for 1 < i 6 X-, there are fewer than m,, vertices having finite degrees, and every vertex having infinite degree has degree m, for some i < k.

Necessary and sufficient conditions for transversals of countable set systems

Abstract This paper proves a conjecture of C. St. J. A. Nash-Williams giving necessary and sufficient conditions for an arbitrary countable system of sets to have a transversal.

A partition relation for triples using a model of Todorccevic

TodorÂ?eviÂ? has shown that there is a ccc extension M in which MAÂ?1 + 2Â? = Â?2 holds and also in which the partition relation Â?i Â? (Â?1,Â?)2 holds for every denumerable ordinal Â?. We show that the partition relation for triples Â?1 Â? (Â?2 + 1, 4)3 holds in the model M, and hence by absoluteness this is a theorem in ZFC.

On directed graphs with an independent covering set

We prove that if a directed graph,D, contains no odd directed cycle and, for all but finitely many vertices, EITHER the in-degrees are finite OR the out-degrees are at most one, thenD contains an independent covering set (i.e. there is a kernel). We also give an example of a countable directed graph which has no directed cycle, each vertex has out-degree at most two, and which has no independent covering set.

The use of elementary substructures in combinatorics

Abstract We illustrate the usefulness of elementary substructures in dealing with problems in infinitary combinatorics.

On a conjecture of Ro¨dl and Voigt

Abstract We prove that if λ is an infinite cardinal number and G is any graph of cardinality κ = λ + which is a union of a finite number of forests, then there is a graph H k of size κ (which does not depend upon G ) so that H κ → ( G ) 1 λ . Rodl and Voight conjectured that there is such a graph H κ for the special case when G is the regular tree on κ in which every vertex has degree κ. We also prove that if a graph is the union of n forests, then it has colouring number 2 n .

The ANTI-order and the fixed point property for caccc posets

This paper settles a conjecture made in (Li 2]) that, if ?P?: ? ? ?? is an ANTI-perfect sequence in a connected caccc poset having no one-way infinite fence, then either there are ? x) and P?(< x) are both fixed point free (fpf), in which case P is also fpf, orP has the fixed point property (fpp) if and only if P? has the fpp.

Isomorphic ANTI-cores of caccc posets

Abstract This paper is a continuation of [1–3]. We show that, for a connected caccc poset having no one-way infinite fence any two ANTI-perfect sequences have the same length and any two ANTI-cores are isomorphic.

A compactness theorem for perfect matchings in matroids

Abstract A family, E , of finite sets is said to be finite matching extendable (f.m.e) if, for every finite matching M ⊆ E and every vertex x ∈ U E ⧹U M , there is E ∈ E such that x ∈ E and E N U M = O. It is shown that, if a matroid is f.m.e., then it has a perfect matching. This generalizes a well-known theorem of Nash-Williams which states that the edges of a graph can be covered by edge-disjoint circuits if every finite set of edges can be so covered.

Independent transversals for countable set systems

Let F=〈Fi¦i∈I〉 be a system of subsets of S and let ℳ be an independence structure on S. A subsystem F K is ℳ-critical if it has an independent transversal versal and whenever B is a transversal of FK in ℳ, then B is a maximal independent subset of F(K)=Ui∈KFi. It is shown that a necessary and sufficient condition for the existence of an independent transversal of a countable system F is that Fi does not depend upon F(K) whenever F K is an ℳ-critical subsystem and i∈I\K.