0
votes
1answer
26 views

T/F prove for modified Ramsey's theorem

By Ramsey's theorem we know that: $\forall k \in \mathbb N : \exists N \in \mathbb N$ that an arbitrary graph $G$ on a set of vertices $\{1,2,...,N\}$ contains $k$ vertices, which represent either a ...
4
votes
0answers
72 views

Arbitrary ratio sequences on a partition of $\mathbb{R}$ (Partition regularity of fixed ratio sequences)

Background: This question arose purely recreationally and doesn't really fit into any context that I know of. Let $A \sqcup B = \mathbb{R}$ be a partition of the ...
2
votes
1answer
173 views

Coloring Vertices of a 50-gon

This is a problem that I have spent a good 2 hours on but can seem to come up with any rigorous solution. If someone could provide one that would be great! If we color 21 vertices of a 50-gon, how do ...