John Ginsburg

Countable spread of exp Y and λY

Abstract We show that it is consistent with ZFC that there exists a compact 0-dimensional Hausdorff space X for which exp X has countable spread, but X is not metrizable. This establishes the independence of Malyhins problem. The space X also has no uncountable weakly separated subspaces, its superextension is first countable, and its square is a strong S -space. For 0-dimensional Y we prove that λ Y has countable spread iff Y is compact and metrizable. We show that it is consistent with ZFC that if Y is 0-dimensional and λ Y is first countable, then Y is compact and metrizable.

Compactness and subsets of ordered sets that meet all maximal chains

LetP be a chain complete ordered set. By considering subsets which meet all maximal chains, we describe conditions which imply that the space of maximal chains ofP is compact. The symbolsP1 andP2 refer to two particular ordered sets considered below. It is shown that the space of maximal chains ℳ (P) is compact ifP satisfies any of the following conditions: (i)P contains no copy ofP1 or its dual and all antichains inP are finite. (ii)P contains no properN and every element ofP belongs to a finite maximal antichain ofP. (iii)P contains no copy ofP1 orP2 and for everyx inP there is a finite subset ofP which is coinitial abovex. We also describe an example of an ordered set which is complete and densely ordered and in which no antichain meets every maximal chain.

A length-width inequality for partially ordered sets with two-element cutsets

Abstract We show that if P is a partially ordered set of width n , and A is an antichain of size n whose elements all have cutsets of size at most 2, then every maximal chain of P has at least n − 2 elements. We also give an extension to larger cutsets.

Ladders in Ordered Sets

An ordered set P is said to have the 2-cutset property if for every element x of P there is a subset S of P whose elements are noncomparable to x, such that |S|≤2 and such that every maximal chain in P meets {x}∪S. It is shown that if P has the 2-cutset property and has width n then P contains a ladder of length [1/2(n−3)].

Chains and discrete sets in zero-dimensional compact spaces

Let X be a compact zero-dimensional space and let B{X) denote the Boolean algebra of all clopen subsets of X. Let k be an infinite cardinal. It is shown that if B( X) contains a chain of cardinality k then X x X contains a discrete subset of cardinality k. This complements a recent result of J. Baumgartner and P. Komjath relating antichains in B(X) to the w-weight of X.

On then-cutset property

For a partially ordered setP and an elementx ofP, a subsetS ofP is called a cutset forx inP if every element ofS is noncomparable tox and every maximal chain ofP meets {x}∪S. We letc(P) denote the smallest integerk such that every elementx ofP has a cutsetS with ‖S‖⩽k: Ifc(P)⩽n we say thatP has then-cutset property. Our results bear on the following question: givenP, what is the smallestn such thatP can be embedded in a partially ordered set having then-cutset property? As usual, 2n denotes the Boolean lattice of all subsets of ann-element set, andBn denotes the set of atoms and co-atoms of 2n. We establish the following results: (i) a characterization, by means of forbidden configurations, of whichP can be embedded in a partially ordered set having the 1-cutset property; (ii) ifP contains a copy of 2n, thenc(P)⩾2[n/2]−1; (iii) for everyn>3 there is a partially ordered setP containing 2n such thatc(P)<c(2n); (iv) for every positive integern there is a positive integerN such that, ifBm is contained in a partially ordered set having then-cutset property, thenm⩽N.

Completely Disconnecting the Complete Graph

In this paper we consider procedures for completely disconnecting a complete graph on

On the dimension of ordered sets with the 2-cutset property

n

On complete partially ordered sets and compatible topologies

vertices

L-spaces in complete spaces of countable tightness using ♢

K_n