Tamás Szőnyi (Eötvös Loránd University, Hungary)

Title: Blocking sets with respect to special substructures of projective planes

Abstract: A substructure $S$ of a projective plane is a set of points $P$ and a set of lines $L$ with the property that that every line $ellin L$ contains at least 2 points of $P$. A blocking set in $S$ is a subset $B$ of $P$ with the property that every line in $L$ contains at least one point of $B$. The aim is to prove (non-trivial) lower bounds on the size of blocking sets. This setting is too general to get interesting results, so we shall consider special substructures. For example the affine plane is such a substructure and for that case we have the famous results of Jamison, Brouwer-Schrijver, when the plane is desarguesian. This shows that one can get interesting results and the small blocking sets are not always related to small blocking sets of the entire projective plane.
Another interesting substructure (again in the desarguesian plane) is to consider exterior (or secant) lines with respect to a given conic. In these cases there are results by Aguglia-Korchm'aros,
Giulietti-Montanucci, Blokhuis-Korchm'aros-Mazzocca.

In the present talk we consider the following substructure: let us
consider a desarguesian plane of order $q^2$ and let $P$ be the union
of the points of $t$ disjoint Baer subplanes. It is well known that every line intersects $P$ in either $t$ or $q+t$ points. Let $L$ be the set of lines meeting $P$ in exactly $q+t$ points. One construction for a blocking set in this substructure is to take one subline in each Baer subplane, another one is to take one of the Baer subplanes. For small $t$, the first construction has size $t(q+1)$ and we show that they are the smallest blocking sets of this substructure if $t$ is less than $sqrt q/2$. For large $t$, the other trivial construction is optimal.

This talk is based on a joint work with Aart Blokhuis and Leo Storme.