✍ Scribed by Chong-Keang Lim
No coin nor oath required. For personal study only.
A graph G is called a supercompact graph if G is the intersection graph of some family 𝒯 of subsets of a set X such that 𝒯 satisfies the Helly property and for any x≠y in X, there exists S ∈ 𝒯 with x ∈ S, y ∉ S. Various characterizations of supercompact graphs are given. It is shown that every clique‐critical graph is supercompact. Furthermore, for any finite graph, H, there is at most a finite number of different supercompact graphs G such that H is the clique‐graph of G.
📜 SIMILAR VOLUMES
De Groojt and 'Verbeek have both asked for an example of a compact Hausdorff space which is not supercompact. It is shown here that if X is nor pseudocompact! then fiX is not supercompact. It is done in the more general setting of Wallman compactificatiom!~.
## Abstract Suppose that κ is indestructibly supercompact and there is a measurable cardinal λ > κ. It then follows that __A__~1~ = {δ < κ∣δ is measurable, δ is not a limit of measurable cardinals, and δ is not δ^+^ supercompact} is unbounded in κ. If in addition λ is 2^λ^ supercompact, then __A__~
## Abstract We determine the large cardinal consistency strength of the existence of a λ‐supercompact cardinal κ such that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {GCH}$\end{document} fails at λ. Indeed, we show that the existence of a λ‐supercompact car
A graph with n edges is called harmonious if it is possible to label the vertices with distinct numbers (modulo n) in such a way that the edge labels which are sums of end-vertex labels are also distinct (modulo n). A ;seneralization of harmonious graphs is felicitous graphs. In felicitous labelling