On the size of a blocking set in \(\text{PG}(2,p)\)
This short and elementary note contains a proof of the following: the size of a non-trivial blocking set in \(\text{PG}(2,p)\), the Desarguesian projective plane of order \(p\), where \(p\) is prime, is at least \(3(p+1)/2\), and this bound can be achieved. (A non-trivial blocking set in \(\text{PG}(2,p)\) is such that any line intersects it, but is not contained in it.) This settles a 25 years old conjecture of Di Paola. This contrasts other results for projective planes of order \(q\). If \(q\) is a square, the tight bound is \(q+ \sqrt q+1\), established by \textit{A. A. Bruen} [SIAM J. Appl. Math. 21, 380-392 (1971; Zbl 0252.05014)]. If the projective plane is Desarguesian, it is at least \(q+ \sqrt{2q}+ 1\), as proved by the author and \textit{A. E. Brouwer} [Bull. Lond. Math. Soc. 18, 132-134 (1986; Zbl 0563.05016)].
scientific article; zbMATH DE number 567475