The non-existence of certain large minimal blocking sets.

*(English)*Zbl 0917.51012The paper under review is concerned with the size of minimal blocking sets. In a projective plane \(\Pi_q\) of order \(q\), a blocking \(k\)-set is a set of \(k\) elements which meets every line but contains no line completely. A blocking set \(\mathcal B\) is called minimal iff for each \(P\in \mathcal B\), \(\mathcal B\setminus\{P\}\) is not a blocking set. A basic problem in Finite Geometry is to find the values of \(k\) for which a minimal blocking \(k\)-set (in a given projective plane) exists. For a nice survey on blocking sets see A. Blokhuis [Bolyai Soc. Math. Stud. 2, 133-155 (1996; Zbl 0849.51005)].

Let \(\mathcal B\) be a minimal blocking \(k\)-set in \(\Pi_q\). It is known that (1) \(q+\sqrt q +1\leq k\leq q\sqrt q +1\); (2) \(k=q+\sqrt q+1\) iff \(q\) is a square and \(\mathcal B\) is a Baer plane; (3) \(k=q\sqrt q +1\) iff \(q\) is a square and \(\mathcal B\) is a unital [see A. A. Bruen, Bull. Am. Math. Soc. 76, 342-344 (1970; Zbl 0207.02601), SIAM J. Appl. Math. 21, 380-392 (1971; Zbl 0252.05014), A. A. Bruen and J. A. Thas, Geom. Dedicata 6, 193-203 (1977; Zbl 0367.05009)]. If, in addition, the plane \(\Pi_q\) is Desarguesian and \(q\geq 16\) is a square, then there do not exist minimal blocking \(k\)-sets with \(q+\sqrt q +2\leq k\leq q+2\sqrt q\) [see S. Ball and A. Blokhuis, Finite Fields Appl. 2, 125-137 (1996; Zbl 0896.51008)]. If \(q=9\) the foregoing holds with \(k=14\), by a result of A. A. Bruen and R. Silverman [Eur. J. Comb. 8, 351-356 (1987; Zbl 0638.51013)]. However it is not known whether or not there exists a minimal blocking 15-set in \(\Pi_9\).

Next step is to look at large minimal blocking \(k\)-sets with \(k\) close to the upper bound \(q\sqrt q +1\). A. Blokhuis and K. Metsch [Math. Soc. Lect. Note. Ser. 191, 37-52 (1993; Zbl 0797.51012)] noticed that there does not exist a blocking \(q\sqrt q\)-set in Desarguesian projective planes of square order \(q\geq 25\).

In the paper under review the author extends this result to an arbitrary projective plane of order 9. The main ingredients of the proof are standard diophantine equations; inner and outer point equations associated to blocking sets; a previous result of the author and A. Maturo concerning blocking sets in \(\Pi_7\) ; and a case by case elimination of 19 possible cases. Finally, the case \(q=16\) of Blokhuis and Metsch’s result remains open and it seems difficult to extend the techniques of this paper to this case.

Let \(\mathcal B\) be a minimal blocking \(k\)-set in \(\Pi_q\). It is known that (1) \(q+\sqrt q +1\leq k\leq q\sqrt q +1\); (2) \(k=q+\sqrt q+1\) iff \(q\) is a square and \(\mathcal B\) is a Baer plane; (3) \(k=q\sqrt q +1\) iff \(q\) is a square and \(\mathcal B\) is a unital [see A. A. Bruen, Bull. Am. Math. Soc. 76, 342-344 (1970; Zbl 0207.02601), SIAM J. Appl. Math. 21, 380-392 (1971; Zbl 0252.05014), A. A. Bruen and J. A. Thas, Geom. Dedicata 6, 193-203 (1977; Zbl 0367.05009)]. If, in addition, the plane \(\Pi_q\) is Desarguesian and \(q\geq 16\) is a square, then there do not exist minimal blocking \(k\)-sets with \(q+\sqrt q +2\leq k\leq q+2\sqrt q\) [see S. Ball and A. Blokhuis, Finite Fields Appl. 2, 125-137 (1996; Zbl 0896.51008)]. If \(q=9\) the foregoing holds with \(k=14\), by a result of A. A. Bruen and R. Silverman [Eur. J. Comb. 8, 351-356 (1987; Zbl 0638.51013)]. However it is not known whether or not there exists a minimal blocking 15-set in \(\Pi_9\).

Next step is to look at large minimal blocking \(k\)-sets with \(k\) close to the upper bound \(q\sqrt q +1\). A. Blokhuis and K. Metsch [Math. Soc. Lect. Note. Ser. 191, 37-52 (1993; Zbl 0797.51012)] noticed that there does not exist a blocking \(q\sqrt q\)-set in Desarguesian projective planes of square order \(q\geq 25\).

In the paper under review the author extends this result to an arbitrary projective plane of order 9. The main ingredients of the proof are standard diophantine equations; inner and outer point equations associated to blocking sets; a previous result of the author and A. Maturo concerning blocking sets in \(\Pi_7\) ; and a case by case elimination of 19 possible cases. Finally, the case \(q=16\) of Blokhuis and Metsch’s result remains open and it seems difficult to extend the techniques of this paper to this case.

Reviewer: Fernando Torres (Campinas)