Reading "The fun of thinking matrix67 math Notes"

Today is the first time I have heard of this story.

In 1933, Hungarian mathematician George Szekeres was only 22 years old. At that time, he often discussed maths with his friends in Budapest, the capital of Hungary. This group of people also have the same born in Hungary, the math geek--paul erd?s great God. But at the time, Erd?s was only 20 years old.

At a math party, a beautiful classmate called Esther Klein put forward a conclusion: on the plane to draw five random points (any three points not collinear), then there must be four points, they constitute a convex quadrilateral. Szekeres and Erd?s and other people thought for a while, did not think how to prove. So, the beautiful classmate proudly announced her proof: the five-point convex hull (covering the entire point set of the smallest convex polygon) can only be Pentagon, quadrilateral and triangular. The first two cases are no longer discussed, and for the third case, the triangle within the two points into a straight line, the triangle of the three vertices must have two vertices in the line of the same side, the four points will constitute a convex quadrilateral.

     the crowd shouted wonderful. After that, Erd?s and Szekeres still dwell on the problem and try to promote it. In the end, they published their paper in 1935, successfully proving a stronger conclusion: for any positive integer n≥3, there is always a positive integer m, so that as long as the points on the plane have m (and any three points are not collinear), a convex N-shaped shape will be found. Erd?s named the question "Happy ending" (Happy ending problem), because the problem brought George Szekeres and the beauty classmate Esther Klein together, and the two were married in 1936.

For a given n, it is advisable to write the minimum required number of points as f (n). Finding the exact value of f (n) is a very small challenge. Since any non-collinear three points on the plane can determine a triangle, f (3) = 3. Esther Klein's conclusion can be simply expressed as f (4) = 5.
When n = 5 o'clock, eight points is not enough. Is eight points without a convex pentagon.

Using some slightly more complex methods, it can be proved that any nine points contain a convex pentagon, so f (5) equals 9.

In 2006, with the help of computers, people finally proved that f (6) = 17. For larger n, what is the value of f (n), respectively? F (n) is there an exact expression? This is one of the problems in mathematics that is not solved.

Anyway, the final result is really happy. In the past 70 years after the marriage, they have been to Shanghai and Adelaide, and eventually settled in Sydney, never separated. August 28, 2005, George and Esther have left the world, a difference of less than one hours.

Reading "The fun of thinking matrix67 math Notes"