MHB The Chromatic Number of the Plane is at Least 5

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The discussion centers on the chromatic number of the plane, which is proposed to be at least 5, contrasting it with the 4-color theorem that applies to planar graphs. An amateur's findings, which have been refined by professionals, suggest that 4 colors are insufficient for certain configurations in the plane. The 4-color theorem is limited to coherent neighboring regions, while this new theorem addresses a broader range of graphs. Participants express curiosity about the differences between the two concepts and the implications of the findings. The conclusion drawn is that at least 5 colors are necessary for proper graph coloring in this context.
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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.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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