AbstractThe Happy End Theorem of Erd˝os and Szekeres asserts that for every integer ngreater than two there is an integer N such that every set of N points in generalposition in the plane includes the n vertices of a convex n-gon. We generalize thistheorem in the framework of certain simple structures, which we call “happy endspaces”.In the winter of 1932/33, Esther Klein observed thatfrom any set of five points in the plane of which no three lie on the same lineit is always possible to select four points that are vertices of a convex polygon.hen she shared this news with a circle of her friends in Budapest, the following prospect generalizing it emerged:Can we find for each integer n greater than two an integer N(n) such thatfrom any set of N(n) points in the plane of which no three lie on the same lineit is always possible to select n points that are vertices of a convex polygon?ndre Makai proved...