Large Incidence-free Sets in Geometries

Large Incidence-free Sets in Geometries

Consider a projective plane of order q with point set P and line set L. The largest value of |X||Y | where X ⊂ P and Y ⊂ L are sets such that no point of X belongs to any line of Y was determined by Haemers and turns out to be q(q −√q + 1)2. This type of problem is interesting for a large class of geometries. More specifically, we investigate this problem for k-dimensional spaces in n-dimension...

LARGE SUM-FREE SETS IN Z/pZ

We show that if p is prime and A is a sum-free subset of Z/pZ with n := |A| > 0.33p, then A is contained in a dilation of the interval [n, p−n] (mod p).

Blocking sets in chain geometries

Quotients of incidence geometries

We develop a theory for quotients of geometries and obtain sufficient conditions for the quotient of a geometry to be a geometry. These conditions are compared with earlier work on quotients, in particular by Pasini[11] and Tits[15]. MSC2000 : 05B25, 51E24, 20B25.

On Saturating Sets in Small Projective Geometries

A set of points, S ⊆ PG(r, q), is said to be %-saturating if, for any point x ∈ PG(r, q), there exist %+ 1 points in S that generate a subspace in which x lies. The cardinality of a smallest possible set S with this property is denoted by k(r, q, %). We give a short survey of what is known about k(r, q, 1) and present new results for k(r, q, 2) for small values of r and q. One construction pres...