Combinatorics » Ramsey Theory

Concavity of van der Waerden numbers ★★

Author(s): Landman

For $ k $ and $ \ell $ positive integers, the (mixed) van der Waerden number $ w(k,\ell) $ is the least positive integer $ n $ such that every (red-blue)-coloring of $ [1,n] $ admits either a $ k $-term red arithmetic progression or an $ \ell $-term blue arithmetic progression.

Conjecture   For all $ k $ and $ \ell $ with $ k \geq \ell $, $ w(k,\ell) \geq w(k+1,\ell-1) $.

Keywords: arithmetic progression; van der Waerden

Posted by Bruce Landman
updated June 21st, 2007
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Graph Theory » Coloring » Vertex coloring

Circular coloring triangle-free subcubic planar graphs ★★

Author(s): Ghebleh; Zhu

Problem   Does every triangle-free planar graph of maximum degree three have circular chromatic number at most $ \frac{20}{7} $?

Keywords: circular coloring; planar graph; triangle free

Posted by mdevos
updated June 20th, 2007
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Geometry » Polytopes

Decomposing the truncated octahedron into parallelepipeds ★

Author(s):

Question   Can the truncated octahedron be decomposed into parallelepipeds?

Keywords: decomposition; lattice; parallelepiped

Posted by mdevos
updated June 20th, 2007
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