Some generalizations of Rédei’s theorem
| Autor: | T. Alderson |
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| Rok vydání: | 2006 |
| Předmět: | |
| Zdroj: | Journal of Discrete Mathematical Sciences and Cryptography. 9:97-106 |
| ISSN: | 2169-0065 0972-0529 |
| DOI: | 10.1080/09720529.2006.10698064 |
| Popis: | By the famous theorems of Redei, a set of q points in AG(2, q) (respectively p points in AG(2, p), p prime) is either a line or it determines at least (respectively ) directions. We generalize these results on two fronts. First we provide bounds on the number of directions determined by a set of n≤q points in a general projective plane of order q. Secondly, given a dual n-arc in Π=PG(k, q) we consider Π as embedded in Σ=PG(k+1, q) where E=Σ–Π is the associated affine space. A collection of affine points is a transversal set of if any line incident with a k-fold point of is incident with at most one point of S. We reformulate Redei’s results in the plane as results on transversal sets. In this setting we generalize Redei’s theorems to higher dimensions. We also provide a new proof of a well known theorem on extending arcs in PG(k, q). |
| Databáze: | OpenAIRE |
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