An Inequality for Incidence Matrices

J. R. Isbell
1959 Proceedings of the American Mathematical Society  
This paper establishes an extremal property for finite projective planes; it is a characteristic property, if we disregard configurations containing fewer than four points. A finite plane consists of k2 -k + 1 points, with lines each of which contains only k points; nevertheless every two lines meet. This would not be remarkable if, say, we could define a "line" as any set of k points containing a specified point Po; but we must satisfy the homogeneity condition, that each point is contained in
more » ... int is contained in exactly k lines. The theorem says that k2 -k + 1 is the largest number of points which can be arranged in such a pattern. A similar, almost characteristic, inequality has been established by de Bruijn and Erdos1 [l ]: in a set of n points one cannot find more than n subsets, every two of which have exactly one common point. There are two trivial ways of constructing such a family for any n, but if there are to be four points, no three of which are on a line, then the finite projective planes are the only possibilities. Both theorems generalize2 to the case in which a pair of lines must have more than one common point. The precise results are Theorem 1. Suppose v, k, \are positive integers, and in a set of v points certain distinct subsets are designated as lines, in such a way that (a) Every two lines have at least X common points; (b) The number of lines containing a given point is the same for all points; and (c) No line contains more than k points. Then \(v -l) -^k2 -k. In case of equality, the points and lines form a (v, k, X) configuration. If\= 1 and k>2, this is a finite projective plane. Theorem 2. For v>0 and X2;0, a set of v points contains no family of more than v distinct nonempty subsets the intersection of any two of which consists of exactly X points. In Theorem 2, the (v, k, X) configurations of course furnish examples. For X = 0 (we may extend the usual definition [2] to this trivial case) there are no exceptional cases of equality; for X = l, the excep-
doi:10.2307/2033580 fatcat:sg7js37iubbdjfdwtfrqevm3dm