The Chromatic Number of the Plane is at Least 5

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SUMMARY

The discussion centers on the chromatic number of the plane, which is established to be at least 5. Participants reference the 4-color theorem, which applies to planar graphs, and contrast it with the broader implications of the current topic. The consensus is that while the 4-color theorem suffices for certain cases, it does not apply universally, necessitating at least 5 colors for more complex graph configurations. The amateur's findings, refined by professionals, are documented in a paper available on arXiv.

PREREQUISITES
  • Understanding of the 4-color theorem
  • Familiarity with graph theory concepts
  • Basic knowledge of chromatic numbers
  • Experience with mathematical proofs and theorems
NEXT STEPS
  • Research the implications of the chromatic number in graph theory
  • Study the differences between planar and non-planar graphs
  • Explore advanced topics in combinatorial mathematics
  • Read the paper on arXiv regarding the chromatic number of the plane
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Mathematicians, graph theorists, and students interested in advanced combinatorial concepts and the implications of the chromatic number in various mathematical contexts.

Ackbach
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I love this stuff! An amateur gets this one, though his result is refined by the pros. His paper on arXiv is here.
 
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Ackbach said:
I love this stuff! An amateur gets this one, though his result is refined by the pros. His paper on arXiv is here.
Okay, why isn't the number four? Is this diagram of his not flat in the plane? I messed around with the 4 color theorem as a kid and I heard it was proved at some point. How is this one different?

-Dan
 
topsquark said:
Okay, why isn't the number four? Is this diagram of his not flat in the plane? I messed around with the 4 color theorem as a kid and I heard it was proved at some point. How is this one different?

-Dan

The 4-color theorem is for coherent neighboring countries.
This is a more general theorem for graphs for which apparently 4 colors is not enough. We need at least 5 colors.
 

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