Topology

Distributivity of a lattice of funcoids is not provable without axiom of choice ★

Author(s): Porton

Conjecture   Distributivity of the lattice $ \mathsf{FCD}(A;B) $ of funcoids (for arbitrary sets $ A $ and $ B $) is not provable in ZF (without axiom of choice).

A similar conjecture:

Conjecture   $ a\setminus^{\ast} b = a\#b $ for arbitrary filters $ a $ and $ b $ on a powerset cannot be proved in ZF (without axiom of choice).

Keywords: axiom of choice; distributive lattice; distributivity; funcoid; reverse math; reverse mathematics; ZF; ZFC

Posted by porton
updated August 24th, 2013
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Graph Theory » Basic G.T.

Almost all non-Hamiltonian 3-regular graphs are 1-connected ★★

Author(s): Haythorpe

Conjecture   Denote by $ NH(n) $ the number of non-Hamiltonian 3-regular graphs of size $ 2n $, and similarly denote by $ NHB(n) $ the number of non-Hamiltonian 3-regular 1-connected graphs of size $ 2n $.

Is it true that $ \lim\limits_{n \rightarrow \infty} \displaystyle\frac{NHB(n)}{NH(n)} = 1 $?

Keywords: Hamiltonian, Bridge, 3-regular, 1-connected

Posted by mhaythorpe
updated August 23rd, 2013
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Graph Theory » Coloring » Vertex coloring

Erdős–Faber–Lovász conjecture ★★★

Author(s): Erdos; Faber; Lovasz

Conjecture   If $ G $ is a simple graph which is the union of $ k $ pairwise edge-disjoint complete graphs, each of which has $ k $ vertices, then the chromatic number of $ G $ is $ k $.

Keywords: chromatic number

Posted by Jon Noel
updated August 22nd, 2013
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Number Theory » Analytic N.T.

Are there only finite Fermat Primes? ★★★

Author(s):

Conjecture   A Fermat prime is a Fermat number \[ F_n = 2^{2^n } + 1 \] that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.

Keywords:

Posted by kurtulmehtap
updated August 21st, 2013
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Number Theory » Analytic N.T.

Are all Fermat Numbers square-free? ★★★

Author(s):

Conjecture   Are all Fermat Numbers \[ F_n = 2^{2^{n } } + 1 \] Square-Free?

Keywords:

Posted by kurtulmehtap
updated August 21st, 2013
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