Question

На плоскости даны n точек(никакие три не лежат на одной прямой) соединенные отрезками. каждый отрезок покрашен в один из трех цветов. при каком наименьшем n можно гарантированно найти треугольник с одноцветными сторонами?

Answer 1

При наименьшем 17 треугольников с одноцветными сторонами

Answer 2

Finding a Triangle with Monochromatic Sides

To guarantee finding a triangle with monochromatic sides, we can use the concept of Ramsey numbers. The Ramsey number for a triangle with monochromatic sides is denoted as R(3,3). This represents the minimum number of vertices (points) required to guarantee the existence of a triangle with monochromatic sides when the edges (connections between the points) are colored with three colors.

Ramsey Numbers and Lower Bounds

The lower bound for R(3,3) is 6. This means that with 6 points, we can guarantee the existence of a triangle with monochromatic sides, regardless of how the edges are colored with three colors.

Conclusion

Therefore, the minimum number of points required to guarantee finding a triangle with monochromatic sides is 6.