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We introduce compactness theorems for generalized colorings and derive several particular compactness theorems from them. It is proved that the theorems and many of their consequences are equivalent in ZF set theory to BPI, the Prime Ideal Theorem for Boolean algebras. keywords: generalized graph colorings, compactness, prime ideal theorem MSC: 05C15; 03E25… (More)

- Robert Cowen, Stephen H. Hechler, John W. Kennedy, Arthur Steinberg
- Discrete Mathematics
- 2007

An odd neighborhood transversal of a graph is a set of its vertices that intersects the set of neighbors of each of its vertices in an odd number of elements. In the case of grid graphs this odd number will be either one or three. We characterize those grid graphs that have odd neighborhood transversals.

- Stephen H. Hechler
- Studia Logica
- 2001

- Paul Erdös, Stephen H. Hechler, Paul C. Kainen
- Discrete Mathematics
- 1978

Define a simple graph G to be k-superuniversal iff for any k-element simple graph K and for any full subgraph H of K every full embedding of H into G can be extended to a full embedding of K into G . We prove that for each positive integer k there exist finite k-superuniversal graphs, and we find upper and lower bounds on the smallest such graphs . We also… (More)

Recently, R. M. Stephenson has used the Continuum Hypothesis to construct two Ä-closed, separable regular, first countable, noncompact Hausdorff spaces. We show that the assumption of the Continuum Hypothesis can be removed by replacing a lemma used in the original construction to deal with arbitrary almost-disjoint families by the construction of a… (More)

- Stephen H. Hechler
- Discrete Mathematics
- 1977

- Stephen H. Hechler
- Mathematical systems theory
- 1970

In his paper on subsets of the series of natural numbers N. Lusin [2] defines two sets F, G g co to be orthogonal (F I G) iff F c3 G is finite. A family (henceforth the term family will be used only to denote subsets of the power set of co) is said to be internally orthogonal iff for any two distinct members F, G of ~ we have F 2. G. Two families ~ and ~… (More)

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