Show circle with points coloured red and blue must have monochromatic red equilateral triangle











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Colour each point on a circle of radius $frac{1}{2}$ red or blue, such that the region of blue points has length $1$. Prove that we can inscribe an equilateral triangle in the circle such that all three vertices are red.



I think the Pigeonhole Principle will be involved, but don't quite see how to apply it. The length condition also seems a bit hard to work with, so any hints or suggestions would be much appreciated.






combinatorics discrete-mathematics







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    What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
    – John Hughes
    3 hours ago










  • Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
    – Prasiortle
    3 hours ago















up vote
3
down vote

favorite
1












Colour each point on a circle of radius $frac{1}{2}$ red or blue, such that the region of blue points has length $1$. Prove that we can inscribe an equilateral triangle in the circle such that all three vertices are red.



I think the Pigeonhole Principle will be involved, but don't quite see how to apply it. The length condition also seems a bit hard to work with, so any hints or suggestions would be much appreciated.






combinatorics discrete-mathematics







share|cite|improve this question




















  • 1




    What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
    – John Hughes
    3 hours ago










  • Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
    – Prasiortle
    3 hours ago













up vote
3
down vote

favorite
1









up vote
3
down vote

favorite
1






1





Colour each point on a circle of radius $frac{1}{2}$ red or blue, such that the region of blue points has length $1$. Prove that we can inscribe an equilateral triangle in the circle such that all three vertices are red.



I think the Pigeonhole Principle will be involved, but don't quite see how to apply it. The length condition also seems a bit hard to work with, so any hints or suggestions would be much appreciated.






combinatorics discrete-mathematics







share|cite|improve this question















Colour each point on a circle of radius $frac{1}{2}$ red or blue, such that the region of blue points has length $1$. Prove that we can inscribe an equilateral triangle in the circle such that all three vertices are red.



I think the Pigeonhole Principle will be involved, but don't quite see how to apply it. The length condition also seems a bit hard to work with, so any hints or suggestions would be much appreciated.






combinatorics discrete-mathematics




combinatorics discrete-mathematics






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edited 2 hours ago









Jean Marie

28.3k41848




28.3k41848










asked 3 hours ago









Prasiortle

1025




1025








  • 1




    What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
    – John Hughes
    3 hours ago










  • Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
    – Prasiortle
    3 hours ago














  • 1




    What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
    – John Hughes
    3 hours ago










  • Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
    – Prasiortle
    3 hours ago








1




1




What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
– John Hughes
3 hours ago




What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
– John Hughes
3 hours ago












Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
– Prasiortle
3 hours ago




Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
– Prasiortle
3 hours ago










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Make all the red points that are $frac {2pi}3$ from a blue point blue. The measure of the blue points is now no more than $3$, but the circumference of the circle is $pi$. There is at least $pi-3$ of the circle still colored red and any of the red points is on an all red equilateral triangle.






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    Make all the red points that are $frac {2pi}3$ from a blue point blue. The measure of the blue points is now no more than $3$, but the circumference of the circle is $pi$. There is at least $pi-3$ of the circle still colored red and any of the red points is on an all red equilateral triangle.






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      Make all the red points that are $frac {2pi}3$ from a blue point blue. The measure of the blue points is now no more than $3$, but the circumference of the circle is $pi$. There is at least $pi-3$ of the circle still colored red and any of the red points is on an all red equilateral triangle.






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        up vote
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        up vote
        5
        down vote









        Make all the red points that are $frac {2pi}3$ from a blue point blue. The measure of the blue points is now no more than $3$, but the circumference of the circle is $pi$. There is at least $pi-3$ of the circle still colored red and any of the red points is on an all red equilateral triangle.






        share|cite|improve this answer












        Make all the red points that are $frac {2pi}3$ from a blue point blue. The measure of the blue points is now no more than $3$, but the circumference of the circle is $pi$. There is at least $pi-3$ of the circle still colored red and any of the red points is on an all red equilateral triangle.







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        share|cite|improve this answer










        answered 3 hours ago









        Ross Millikan

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        289k23195367






























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