# Spanning Graphs and the Axiom of Choice

@article{Delhomm2006SpanningGA, title={Spanning Graphs and the Axiom of Choice}, author={Christian Delhomm{\'e} and Marianne Morillon}, journal={Reports Math. Log.}, year={2006}, volume={40}, pages={165-180} }

A b s t r a c t. We show in set-theory ZF that the axiom of choice is equivalent to the statement every bipartite connected graph has a spanning sub-graph omitting some complete finite bipartite graph K n,n , and in particular it is equivalent to the fact that every connected graph has a spanning cycle-free graph (possibly non connected). We consider simple undirected loop-free graphs. A forest is a graph with no cycles, a tree is a connected forest. A graph G is a sub-graph of G if all its…

## 7 Citations

Partition models, Permutations of infinite sets without fixed points, Variants of CAC, and weak forms of AC

- Mathematics
- 2021

We study new relations of the following statements with weak choice principles in ZF (ZermeloFraenkel set theory without the Axiom of Choice (AC)) and ZFA (ZF with the axiom of extensionality…

Linear forms and axioms of choice

- Mathematics
- 2009

We work in set-theory without choice ZF. Given a commutative field $\mathbb K$, we consider the statement $\mathbf D (\mathbb K)$: “On every non null $\mathbb K$-vector space there exists a non-null…

König's Infinity Lemma and Beth's Tree Theorem

- Mathematics
- 2017

König, D. [1926. ‘Sur les correspondances multivoques des ensembles’, Fundamenta Mathematica, 8, 114–34] includes a result subsequently called König's Infinity Lemma. Konig, D. [1927. ‘Über eine…

Commentationes Mathematicae Universitatis Carolinae

- 2013

In set theory without the Axiom of Choice ZF, we prove that for every commutative field K, the following statement DK: “On every non null K-vector space, there exists a non null linear form” implies…

THE PROSPECTS OF UNLIMITED CATEGORY THEORY: DOING WHAT REMAINS TO BE DONE

- Mathematics, Computer ScienceThe Review of Symbolic Logic
- 2015

It is shown that the answer is no by showing that unlimited category theory is inconsistent.

Maximal independent sets, variants of chain/antichain principle and cofinal subsets without AC

- Mathematics
- 2020

In set theory without the Axiom of Choice (AC), we observe new relations of the following statements with weak choice principles. 1. Every locally finite connected graph has a maximal independent…

## References

SHOWING 1-10 OF 12 REFERENCES

THE AXIOM OF CHOICE

- 2003

We propose that failures of the axiom of choice, that is, surjective functions admitting no sections, can be reasonably classified by means of invariants borrowed from algebraic topology. We show…

Equivalents of the Axiom of Choice II

- Mathematics
- 1985

Set Forms. The Well-Ordering Theorem. The Axiom of Choice. The Law of the Trichotomy. Maximal Principles. Forms Equivalent to the Axiom of Choice Under the Axioms of Extensionality and Foundation.…

Equivalents of the axiom of choice

- Mathematics
- 2001

Parts of this note have been discussed elsewhere in the blog, sometimes in a different form (see here and here, for example), but I haven’t examined before the equivalence of 5–7 with choice. In the…

The strength of Mac Lane set theory

- Mathematics, Computer ScienceAnn. Pure Appl. Log.
- 2001

This paper shows that the consistency strength of Mac Lane's system is not increased by adding the axioms of Kripke–Platek set theory and even the Axiom of Constructibility to Mac Lane’s axiOMs, and studies a simple set theoretic assertion.

Some remarks and problems on the colouring of infinite graphs and the theorem of Kuratowski

- Mathematics
- 1964

Consequences of the axiom of choice

- Mathematics
- 1998

Numerical list of forms Topical list of forms Models Notes References for relations between forms Bibliography Table 1 and Table 2 Subject index Author index Software.

Admissible sets and structures

- Mathematics
- 1978

Let's read! We will often find out this sentence everywhere. When still being a kid, mom used to order us to always read, so did the teacher. Some books are fully read in a week and we need the…

Admissible sets and structures, Springer-Verlag, Berlin

- 1975

The Axiom of Choice, Studies in Logic

- The Axiom of Choice, Studies in Logic
- 1973