Show circle with points coloured red and blue must have monochromatic red equilateral triangle
$begingroup$
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
edited Dec 9 '18 at 22:16
Jean Marie
29.6k42050
asked Dec 9 '18 at 21:31
PrasiortlePrasiortle
1525
$endgroup$
add a comment |
$begingroup$
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
edited Dec 9 '18 at 22:16
Jean Marie
29.6k42050
asked Dec 9 '18 at 21:31
PrasiortlePrasiortle
1525
$endgroup$
1
$begingroup$
What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
$endgroup$
– John Hughes
Dec 9 '18 at 21:41
$begingroup$
Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
$endgroup$
– Prasiortle
Dec 9 '18 at 21:53
add a comment |
$begingroup$
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
edited Dec 9 '18 at 22:16
Jean Marie
29.6k42050
asked Dec 9 '18 at 21:31
PrasiortlePrasiortle
1525
$endgroup$
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
edited Dec 9 '18 at 22:16
Jean Marie
29.6k42050
asked Dec 9 '18 at 21:31
PrasiortlePrasiortle
1525
edited Dec 9 '18 at 22:16
Jean Marie
29.6k42050
asked Dec 9 '18 at 21:31
PrasiortlePrasiortle
1525
edited Dec 9 '18 at 22:16
Jean Marie
29.6k42050
edited Dec 9 '18 at 22:16
Jean Marie
29.6k42050
edited Dec 9 '18 at 22:16
Jean Marie
29.6k42050
29.6k42050
asked Dec 9 '18 at 21:31
PrasiortlePrasiortle
1525
asked Dec 9 '18 at 21:31
PrasiortlePrasiortle
1525
asked Dec 9 '18 at 21:31
PrasiortlePrasiortle
1525
1525
1
$begingroup$
What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
$endgroup$
– John Hughes
Dec 9 '18 at 21:41
$begingroup$
Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
$endgroup$
– Prasiortle
Dec 9 '18 at 21:53
add a comment |
1
$begingroup$
What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
$endgroup$
– John Hughes
Dec 9 '18 at 21:41
$begingroup$
Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
$endgroup$
– Prasiortle
Dec 9 '18 at 21:53
1
1
$begingroup$
What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
$endgroup$
– John Hughes
Dec 9 '18 at 21:41
$begingroup$
What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
$endgroup$
– John Hughes
Dec 9 '18 at 21:41
$begingroup$
Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
$endgroup$
– Prasiortle
Dec 9 '18 at 21:53
$begingroup$
Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
$endgroup$
– Prasiortle
Dec 9 '18 at 21:53
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Make all the red points that are a distance exactly $frac {2pi}3$ away 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.
edited Dec 10 '18 at 5:22
Acccumulation
6,9042618
answered Dec 9 '18 at 21:41
Ross MillikanRoss Millikan
295k23198371
$endgroup$
add a comment |
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Make all the red points that are a distance exactly $frac {2pi}3$ away 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.
edited Dec 10 '18 at 5:22
Acccumulation
6,9042618
answered Dec 9 '18 at 21:41
Ross MillikanRoss Millikan
295k23198371
$endgroup$
add a comment |
$begingroup$
Make all the red points that are a distance exactly $frac {2pi}3$ away 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.
edited Dec 10 '18 at 5:22
Acccumulation
6,9042618
answered Dec 9 '18 at 21:41
Ross MillikanRoss Millikan
295k23198371
$endgroup$
add a comment |
$begingroup$
Make all the red points that are a distance exactly $frac {2pi}3$ away 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.
edited Dec 10 '18 at 5:22
Acccumulation
6,9042618
answered Dec 9 '18 at 21:41
Ross MillikanRoss Millikan
295k23198371
$endgroup$
Make all the red points that are a distance exactly $frac {2pi}3$ away 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.
edited Dec 10 '18 at 5:22
Acccumulation
6,9042618
answered Dec 9 '18 at 21:41
Ross MillikanRoss Millikan
295k23198371
edited Dec 10 '18 at 5:22
Acccumulation
6,9042618
edited Dec 10 '18 at 5:22
Acccumulation
6,9042618
edited Dec 10 '18 at 5:22
Acccumulation
6,9042618
6,9042618
answered Dec 9 '18 at 21:41
Ross MillikanRoss Millikan
295k23198371
answered Dec 9 '18 at 21:41
Ross MillikanRoss Millikan
295k23198371
answered Dec 9 '18 at 21:41
Ross MillikanRoss Millikan
295k23198371
295k23198371
add a comment |
add a comment |
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$begingroup$
What does "region of blue points has length 1" mean? Is one of the assumptions that the blue set is measurable?
$endgroup$
– John Hughes
Dec 9 '18 at 21:41
$begingroup$
Yes, we are essentially assuming that you can get all the blue points together in a line and measure its length.
$endgroup$
– Prasiortle
Dec 9 '18 at 21:53