In a graph $\Gamma=(V,E)$ a vertex $v$ is resolved by a vertex-set $S=\{v_1,\ldots,v_n\}$ if its (ordered) distance list with respect to $S$, $(d(v,v_1),\ldots,d(v,v_n))$, is unique. A set $A\subset V$ is resolved by $S$ if all its elements are resolved by $S$. $S$ is a resolving set in $\Gamma$ if it resolves $V$. The metric dimension of $\Gamma$ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving set is a set of vertices in one of the vertex classes that resolves the other class.We show that the metric dimension of the incidence graph of a finite projective plane of order $q\geq 23$ is $4q-4$, and describe all resolving sets of that size. Let $\tau_2$ denote the size of the smallest double blocking set in PG$(2,q)$, the Desarguesian projective plane of order $q$. We prove that for a semi-resolving set $S$ in the incidence graph of PG$(2,q)$, $|S|\geq \min \{2q+q/4-3, \tau_2-2\}$ holds. In particular, if $q\geq9$ is a square, then the smallest semi-resolving set in PG$(2,q)$ has size $2q+2\sqrt{q}$. As a corollary, we get that a blocking semioval in PG$(2, q)$, $q\geq 4$, has at least $9q/4-3$ points. A corrigendum was added to this paper on March 3, 2017.

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Starting with a result in combinatorial number theory we prove that (apart from a couple of exceptions that can be classified), for any elements a 1 , . . . ,a n of GF(q), there are distinct field elements b 1 , . . . , b n such that a 1 b 1 +· · ·+a n b n = 0. This implies the classification of hyperplanes lying in the union of the hyperplanes X i = X j in a vector space over GF(q), and also the classification of those multisets for which all reduced polynomials of this range are of reduced degree q − 2. The proof is based on the polynomial method.

The main purpose of this paper is to find double blocking sets in PG(2, q) of size less than 3q, in particular when q is prime. To this end, we study double blocking sets in PG(2, q) of size 3q − 1 admitting at least two (q − 1)-secants. We derive some structural properties of these and show that they cannot have three (q − 1)-secants. This yields that one cannot remove six points from a triangle, a double blocking set of size 3q, and add five new points so that the resulting set is also a double blocking set. Furthermore, we give constructions of minimal double blocking sets of size 3q − 1 in PG(2, q) for q = 13, 16, 19, 25, 27, 31, 37 and 43. If q > 13 is a prime, these are the first examples of double blocking sets of size less than 3q. These results resolve two conjectures of Raymond Hill from 1984.

A 2-fold blocking set (double blocking set) in a finite projective plane Π is a set of points, intersecting every line in at least two points. The minimum number of points in a double blocking set of Π is denoted by τ 2 (Π). Let PG(2, q) be the Desarguesian projective plane over GF(q), the finite field of q elements. We show that if q is odd, not a prime, and r is the order of the largest proper subfield of GF(q), then τ 2 (PG(2, q)) ≤ 2(q + (q − 1)/(r − 1)).For a finite projective plane Π, letχ(Π) denote the maximum number of classes in a partition of the point-set, such that each line has at least two points in some partition class. It can easily be seen thatχ(Π) ≥ v − τ 2 (Π) + 1 (⋆) for every plane Π on v points. Let q = p h , p prime. We prove that for Π = PG(2, q), equality holds in (⋆) if q and p are large enough.