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.

We consider the following q-analog of the basic combinatorial search problem: let q be a prime power and GF(q) the finite field of q elements. Let V denote an ndimensional vector space over GF(q) and let v be an unknown 1-dimensional subspace of V . We will be interested in determining the minimum number of queries that is needed to find v provided all queries are subspaces of V and the answer to a query U is YES if v U and NO if v U . This number will be denoted by A(n, q) in the adaptive case (when for each queries answers are obtained immediately and later queries might depend on previous answers) and M (n, q) in the non-adaptive case (when all queries must be made in advance).In the case n = 3 we prove 2q − 1 = A(3, q) < M (3, q) if q is large enough. While for general values of n and q we establish the bounds n log q ≤ A(n, q) ≤ (1 + o(1))nq and(1 − o(1))nq ≤ M (n, q) ≤ 2nq, provided q tends to infinity.

In this article we prove a theorem about the number of directions determined by less then q affine points, similar to the result of Blokhuis et. al.[3] on the number of directions determined by q affine points.

Given a set T ⊆ GF(q), |T | = t, w T is defined as the smallest positive integer k for which y∈T y k = 0. It can be shown that w T t always and w T t − 1 if the characteristic p divides t. T is called a Vandermonde set if w T t − 1 and a super-Vandermonde set if w T = t. This (extremal) algebraic property is interesting for its own right, but the original motivation comes from finite geometries. In this paper we classify small and large super-Vandermonde sets.

In this paper the metric dimension of (the incidence graphs of) particular partial linear spaces is considered. We prove that the metric dimension of an affine plane of order q ≥ 13 is 3q − 4 and describe all resolving sets of that size if q ≥ 23. The metric dimension of a biaffine plane (also called a flag-type elliptic semiplane) of order q ≥ 4 is shown to fall between 2q −2 and 3q −6, while for Desarguesian biaffine planes the lower bound is improved to 8q/3 − 7 under q ≥ 7, and to 3q − 9 √ q under certain stronger restrictions on q. We determine the metric dimension of generalized quadrangles of order (s, 1), s arbitrary. We derive that the metric dimension of generalized quadrangles of order (q, q), q ≥ 2, is at least max{6q − 27, 4q − 7}, while for the classical generalized quadrangles W (q) and Q(4, q) it is at most 8q.