Happy End Problem


The happy end problem, also called the "happy ending problem," is the problem of determining for n>=3 the smallest number of points g(n) in general position in the plane (i.e., no three of which are collinear), such that every possible arrangement of g(n) points will always contain at least one set of n points that are the vertices of a convex polygon of n sides. The problem was so-named by Erdős when two investigators who first worked on the problem, Ester Klein and George Szekeres, became engaged and subsequently married (Hoffman 1998, p. 76).

Since three noncollinear points always determine a triangle, g(3)=3.


Random arrangements of n=4 points are illustrated above. Note that no convex quadrilaterals are possible for the arrangements shown in the fifth and eighth figures above, so g(4) must be greater than 4. E. Klein proved that g(4)=5 by showing that any arrangement of five points must fall into one of the three cases (left top figure; Hoffman 1998, pp. 75-76).


Random arrangements of n=8 points are illustrated above. Note that no convex pentagons are possible for the arrangement shown in the fifth figure above, so g(5) must be greater than 8. E. Makai proved g(5)=9 after demonstrating that a counterexample could be found for eight points (right top figure; Hoffman 1998, pp. 75-76).

As the number of points n increases, the number of k-subsets of n that must be examined to see if they form convex k-gons increases as (n; k), so combinatorial explosion prevents cases much bigger than n=5 from being easily studied. Furthermore, the parameter space become so large that searching for a counterexample at random even for the case n=6 with k=12 points takes an extremely long time. For these reasons, the general problem remains open.

g(6)=17 was demonstrated by Szekeres and Peters (2006) using a computer search which eliminated all possible configurations of 17 points which lacked convex hexagons while examining only a tiny fraction of all configurations.

Erdős and Szekeres (1935) showed that g(n) always exists and derived the bound

 2^(n-2)+1<=g(n)<=(2n-4; n-2)+1,

where (n; k) is a binomial coefficient. For n>=4, this has since been reduced to g(n)<=g_1(n) for

 g_1(n)=(2n-4; n-2)

by Chung and Graham (1998), g(n)<=g_2(n) for

 g_2(n)=(2n-4; n-2)+7-2n

by Kleitman and Pachter (1998), and g(n)<=g_3(n) for

 g_3(n)=(2n-5; n-2)+2

by Tóth and Valtr (1998). For g(6), these bounds give 71, 70, 65, and 37, respectively (Hoffman 1998, p. 78).

The values of g_3(n) for n=6, 7, ... are 37, 128, 464, 1718, ... (OEIS A052473).

See also

Convex Hull, Convex Polygon

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Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, p. 78, 2003.Chung, F. R. K. and Graham, R. L. "Forced Convex n-gons in the Plane." Discr. Comput. Geom. 19, 367-371, 1998.Erdős, P. and Szekeres, G. "A Combinatorial Problem in Geometry." Compositio Math. 2, 463-470, 1935.Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. New York: Hyperion, pp. 75-78, 1998.Kleitman, D. and Pachter, L. "Finding Convex Sets among Points in the Plane." Discr. Comput. Geom. 19, 405-410, 1998.Lovász, L.; Pelikán, J.; and Vesztergombi, K. Discrete Mathematics, Elementary and Beyond. New York: Springer-Verlag, 2003.Sloane, N. J. A. Sequence A052473 in "The On-Line Encyclopedia of Integer Sequences."Szekeres, G. and Peters, L. "Computer Solution to the 17-Point Erdős-Szekeres Problem." ANZIAM J. 48, 151-164, 2006.Tóth, G. and Valtr, P. "Note on the Erdős-Szekeres Theorem." Discr. Comput. Geom. 19, 457-459, 1998.

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Happy End Problem

Cite this as:

Weisstein, Eric W. "Happy End Problem." From MathWorld--A Wolfram Web Resource.

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