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Results tagged with hypergraph
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user 5903
Hypergraphs are generalizations of graphs, where edges can be made of more than two vertices.
1
vote
Property ${\bf B}$ for families of large sets with small intersection
To confirm Andreas' suspicion: Balcar and Vojtáš proved in Almost Disjoint Refinement of families of subsets of $\mathbb{N}$ that every ultrafilter on $\mathbb{N}$ has an almost disjoint refinement. T …
- 11.4k
2
votes
Accepted
Large chromatic number in hypergraphs with large edges
For $\kappa=\aleph_0$ yes: there are (many) models with ultrafilters of character less than $\mathfrak{c}$. Let $E\subseteq[\omega]^\omega$ be a base for an ultrafilter, say $|E|=\aleph_1<\mathfrak{c} …
- 11.4k
4
votes
Accepted
Posets such that the collection of principal down-sets does not have property ${\bf B}$
Let $M$ be the ordered Mostowski model (T. Jech, The Axiom of Choice, Section 4.5). Its set of atoms, $A$, has a linear order $\prec$ that makes it isomorphic to the rationals. Let $S\in M$ be a subse …
- 11.4k
3
votes
Posets such that the collection of principal down-sets does not have property ${\bf B}$
The axiom of choice implies that for every partial order $P$ the
hypergraph $H_P$ has property $B$.
Let $(P,\le)$ be a partial order. …
- 11.4k