Combinatorial and Geometric Structures and Their by A. Barlotti

By A. Barlotti

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Thas, Construction o f maximal arcs and p a r t i a l geometries, Geomet r i a e Dedicata, 3(1974) 61-64. Thas, Construction o f maximal arcs and dual ovals i n t r a n s l a t i o n planes , Europ. J Combi n a t o r i cs , (1 980) 1 , 189-1 92. A. Thas, F. C. Bose t o p a r t i a l geometries, Rend. Acc. Naz. L i n c e i , (8) 59 (1975) 86-90. A.

171 M. T a l l i n i Scafati , Sui {k,n)-archi d i un piano g r a f i c o f i n i t o , Rend. Acc. Naz. L i n c e i , (8) 40 (1966) 1-6. [ 181 M. T a l l i n i S c a f a t i , Sui {k,n)-archi d i un piano g r a f i c o f i n i t o con part i c o l a r e riguardo a q u e l l i con due c a r a t t e r i , Rend. Acc. Naz. L i n c e i , (8) 40 (1966) 812-818, 1020-1025. , 26 (1967) 273-303. [ 191 M. T a l l i n i Scafati un S [ 201 , Rend. r,q M. T a l l i n i S c a f a t i , C a l o t t e d i t i p o (m,n) i n uno spazio d i Galois SrYq, Rend.

2. THE k-SETS WITH TWO CHARACTERS I N PG (r,q) ( r 3 3 ) L e t us examine the known r e s u l t s about the k-sets o f k i n d (man), w i t h 0 3. We proved ( i n [ 191, p r o p o s i t i o n V I I I ) t h a t : Proposition I V . I n PG(r,q) ( r 3 3 ) a k-set o f k i n d (0,n) e x i s t s o n l y i f i t i s e i t h e r n = l o r n = q and then i t i s e i t h e r K { P I o r K=PG(r,q)-PG(r-1,q). S i m i l a r l y (considering the complements) a k-set K o f k i n d (m,qtl) e x i s t s o n l y i f i t i s e i t h e r m = l o r m = q and then i t i s e i t h e r K=PG(r-1,q) - o r K=PG(r,q)- {PI.

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