Parties équitables d'espaces linéaires

Résumé En

. — A subset of a linear space L (we don't mean a vector space) is said to be equitable of ordre n (a cardinal which is not necessarily finite) if its intersection with every line of L has cardinality n. Our main results are : a) Every finite linear space L contains at most one non trivial equitable subset. If E is such a subset of L, then the same number v of lines go through all the points of E, the same number w of lines go through all the points of L \ E and v — w divides v — 1 . b) For every integer e > 6, there is a linear space of cardinality e which contains a non trivial equitable subset. c) Let L be an infinite linear space wherein the lines and the set of lines have the same cardinality d. For every cardinal n ⩾ d, there exist in L equitable subsets of order n. d) For every natural number n, there exist infinite linear spaces with the following property : if the point p does not belong to the line D, there are exactly n lines through p which do not cut D.