of the implicit constants. This result can be used to prove Beck's theorem. A similar bound for the number of incidences is conjectured for point-circle incidences, but only weaker upper bounds are known. Beck's theorem Beck's theorem says that finite collections of points in the plane fall into one of two extremes; one where a large fraction of points lie on a single line, and one where a large number of lines are needed to connect all the points. The theorem asserts the existence of positive constants such that given any points in the plane, at least one of the following statements is true: There is a line that contains at least of the points. There exist at least lines, each of which contains at least two of the points. In Beck's original argument, is 100 and is an unspecified constant; it is not known what the optimal values of and are. More examples Projective geometries Moufang polygon Schläfli double six Reye configuration Cremona–Richmond configuration Kummer configuration Klein configuration Non-Desarguesian planes See also Combinatorial designs Finite geometry Intersection theorem Levi graph Notes References . . . External links incidence system at the Encyclopedia of Mathematics