Extended abstract of a paper presented at Microscopy and Microanalysis 2012 in Phoenix, Arizona, USA, July 29 – August 2, 2012.

Surface and bulk interactions between a metal M (M = Au, Cu or Ni) and monocrystalline α-SiC are studied. Surface interactions are quantified by the work of adhesion measured by the sessile drop technique under high vacuum. Products of M-SiC bulk reactions are characterised by electron microscopy and microprobe analysis. Interpretation of experimental results is based on classical thermodynamics for equilibria of bulk phases and on interfacial thermodynamics for adsorption phenomena and wetting.

(i) J. Baumgartner has kindly drawn our attention to the fact that Theorem 2 as stated in (1) is false. A counter example is the case in which m = ℵ2; n = ℵ1; p = ℵ0. For by reference (3) of the paper (1) there is an almost disjoint family (Aγ: γ < ω1) of infinite subsets of ω̲ Put Aν = ω̲ for ω1 ≤ ν < ω2. Then, contrary to the assertion of that theorem, all conditions of Theorem 2 are satisfied. However, Theorem 2 becomes correct if the hypothesis

is strengthened to

In fact, Baumgartner has proved the desired conclusion under the weaker hypothesis

For a given index set I, let us consider a family (Aν,: ν ∈ I) of subsets of a set E. In this note we deal with some aspects of the following question: to what extent is it possible to prescribe the cardinalities, or the order types in case E is ordered, of the sets Aν and of their pairwise intersections? In (1) the authors have shown that, given any regular cardinal a, there is a family of a+ sets of cardinal a whose pairwise intersections are arbitrarily prescribed to be either less than or equal to a. In Theorem 1 below we prove a stronger result which states that if a is regular, say a = ℵα, and if E is well-ordered and of order type , then one can find a+ subsets Aν, of E, each of type , whose pairwise intersections are arbitrarily prescribed to be either of type ωα or of a type less than ωα. By way of contrast, Theorem 2 below implies – this is its special case m = ℵω; n = ℵ2; p = ℵ0 – that, assuming the Generalized Continuum Hypothesis (GCH), there do not exist ℵω+1 sets Aν, each of cardinal at most ℵω such that ℵ2 of them have pairwise finite intersections, whereas all other pairs of sets Aν have a denumerable intersection. Theorem 3 gives another case in which some type of prescription of the sizes of the intersections cannot be satisfied. Finally, Theorem 4 asserts that in Theorem 3 the condition cfp ≠ cfm cannot be omitted. The paper concludes with some remarks on open questions.

A system or family (Aγ: γ∈ N) of sets Aγ, indexed by the elements of a set N, is called an (a, b)-system if ¦N¦ = a and ¦Aγ¦ = b for γ ∈ N. Expressions such as “(a, <b)-system” are self-explanatory. The system (Aγ: γ∈N) is called a δ-system [1] if Aμ∩Aγ = Ap ∩ Aσ whenever μ, γ, ρ, σ ∈ N; μ≠ γ; ρ ≠ σ. If we want to indicate the cardinality ¦N¦ of the index set N then we speak of a δ(¦N¦) system. In [1] conditions on cardinals a, b, c were obtained which imply that every (a, b)-system contains a δ(c)-subsystem. In [2], for every choice of cardinals b, c such that the least cardinal a = fδ(b, c) was determined which has the property that every (a, < b)-system contains a δ(c)-subsystem.