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Results tagged with infinite-combinatorics
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user 97426
Combinatorial properties of infinite sets. This is a corner-point of set theory and combinatorics.
5
votes
When is it true that if $G$ is isomorphic to a spanning subgraph of $H$ and vice versa, then...
This is an extended comment rather than an actual answer.
I think that any answer to your questions 1 and 2 is likely to be rather involved since the properties you ask are sensitive to small local ch …
- 2,671
4
votes
Accepted
$\omega$-Hamilton paths in $\mathbb{Z}^n$
In "E. Vazonyi, Über Gitterpunkte des mehrdimensionalen Raumes, Acta Litt. Sci. Szeged 9, 163-173 (1939).", it is shown that there is such a path for every $n$ (as well as a Hamilton double ray, i.e. …
- 2,671
4
votes
Accepted
Avoiding multiply covered vertices in graph edge coverings
For $n \in \mathbb N$ Let $a_n, b_n$ be a pair of vertices connected by an edge. For every finite subset $M \subset \mathbb N$ take an additional vertex $v_M$ and connect it to every vertex $a_n$ With …
- 2,671
4
votes
Matching number in infinite hypergraphs
I think if all edges are finite, then there is such a matching.
Let $M_0$ be a maximal matching (in the sense that no edge can be added to it; this exists by Zorn's Lemma). Let $\alpha_0$ be the cardi …
- 2,671
3
votes
Representability of "large" graphs by "small" intersection graphs
You probably want a stronger condition on $G$ than just the number of components:
Let $G$ be a star with $2^\kappa$ leaves. Then the leaves form an independent set of size $2^\kappa$, but there is no …
- 2,671
1
vote
Accepted
Infinite strongly rigid graphs
Unfortunately, I don't have enough reputation to comment, but there seems to be a problem with both solutions suggested so far: The graphs are bipartite, meaning that they allow a homomorphism to a si …
- 2,671