A New Blocking Semioval

Abstract

Let I1 = ( P, L) be a projective plane of order n . A blocking set in IT is a set B o f points such that fo r every line 1 of fl there is at least one point of 1 in B , but 1 is not entirely contained in B . Blocking sets have been extensively studied, see for example, Berardi and Eugeni [2]. A semioval in II is a set S of points such that for every point P E S t here is a unique tangent to S containing P . Here, as usual, a tangent to S is a line of IT meeting S in exactly one point. The concept of semioval is a generalization of the concept of oval. An oval in IT is a set of n + 1 points such that no three are collinear. Since two points in IT lie on a unique line, and since there are n + 1 lines through a point of fl, it is clear t hat an oval is a semioval. Ovals have also been extensively studied, but semiovals have so far received little attention . (See Hughes and Piper [5], Chapter XII.) One type of semioval that has recently received some attention is the blocking semioval. A blocking semioval in TI is a blocking set that is also a semioval. That is, a blocking semioval is a set S of points in II satisfying: (1) e very line 1 of IT contains a point of Sand a point not in S ; (2) fo r every point P of S there is a unique tangent to S containing P . One interesting aspect o f a blocking semioval is that it is both a minimal blocking set and a maximal semioval [4].