Combinatorial and Geometric Structures and Their by A. Barlotti

By A. Barlotti

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Naz. 976) 782788. A. Thas, A combinatorial problem, Geometriae Dedicata 1 (1973) 236-240. Ann& of Discrete Mathematics 14 (1982) 39-56 0 North-HollandPublishing Company ON k-SETS OF KIND (m,n) OF A FINITE PROJECTIVE OR AFFINE SPACE M. T a l l i n i S c a f a t i I s t i t u t o Matematico "G. Castelnuovo", U n i v e r s i t a d i Roma, I t a l y INTRODUCTION I n the f o l l o w i n g paper, f i r s t we e x p l a i n the r e s u l t s known up t o now about We w i l l h ( q = p , p prime), the theory o f k-sets w i t h two characters i n a Galois space PG(r,q).

13), i f II = h - 1 and p = 2 we absurd 1 < s < 0. 13) i t i s 1 < II < h-2. 8), II h-1 - becomes c2 3qc t 2q(q t 1) = 0, which must have i n t e g e r roots, whence A = q(q-8) must be a square and i t happens o n l y when q = 9. 5. k-SETS OF KIND(m,n) I N AN AFFINE SPACE AG(r,q), WITH r 3 3, < q.

Then a l s o the case (11) i s impossible. L e t us now examine the case (111). x+q = 0, being 2X = 2h-2&, we have M. 17) i n the unknown x must have t h e i n t e g e r s o l u t i o n 1t2s. I t f o l l o w s t h a t i t s d i s c r i m i n a n t A must be an i n t e g e r square. 18) whence we i n f e r 2h >h (being A > 0) and moreover h 2X-h A=p( p - 1 ) . 19) If2X = h, we have A = 0; from (111) we have L! 20) (61 ) / 2, m = (q-~;i)/2, n = (qtJ{)/2, whence i t f o l l o w s t h a t q i s an odd square.