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Congressus Numerantium December Google Scholar. Lecture Notes. Preliminary Version July Google Scholar. Kou, Y. Lin, S. In: Proceedings of the ForneyFest. Kluwer Academic, Boston Google Scholar. Li, W. Lubotzky, A.

Margulis, G. Reingold, O. Richardson, T. Rosenthal, J. In: Allerton Conference Google Scholar. Roth, R. Sarnak, P. Shokrollahi, A.

pusrimendranrost.tk In: Reichel, H. STACS Sipser, M. Codes and complexity. Spielman, D. Tanner, R. SIAM J.

Zemor, G. Personalised recommendations. Cite paper How to cite? ENW EndNote. In spite of this she helped me run things smoothly thus making a great though indirect contribution to the success of the Conference. I am thankful to the authors for the many excellent papers in this volume, and also to the referees for their help.

As in the earlier conferences, the various local arrangements were made by the C. Department of Conferences and Institutes. This time, unfortunately, some participants suffered inconveniences. I wish to apologize for the same. I am thankful to North Holland particularly, the desk editor Aad Thoen for their promptness in handling the manuscripts, and producing this volume. Last, but not the least, my thanks go to all the participants in the Symposium for it was their participation which truly made it a success.

DEZAand N.

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Introduction Galois geometry, in its broader sense, is the study of the nonlinear sets of points in finite spaces including here also finite spaces over nonfield structures. References on the early history of Galois geometry can be found in [29]. In this survey the main research areas in which Galois geometry may be divided were clearly indicated: a To offer pure geometric interpretations of algebraic and number theoretic properties. Here the solution may follow from algebraic results or from purely combinatorial and geometric methods.

In the last 20 years the study of Galois geometry has developed in a quite remarkable way. In in some lectures we presented a brief survey on this field see [2, n. We shall exhibit here some of the progress done since , particularly in the above section d , and some old and new open problems. A k; n -arc of r q is a set of k points of r q such that y1 is the largest number of them which are collinear.

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The k ; 2 -arcs are simply called k-arcs. Barlotti 2 is the largest number of them which are collinear. The k ; 2 -caps are simply called k-caps. A line g is an s-secant of a k; n -arc, K, if g contains s points of K ; the 1-secants and 0-secants will be called respectively tangent and external lines to K.

Let t, denote the total number of s-secants to K ; the numbers t, are called the characters of K, and K is said to have p characters if exactly p among its characters are different from zero. The arc K is of type sl, s 2 ,. A generalization of a theorem by Buekenhout In Buekenhout proved the following theorem see [lo] : Theorem.

This theorem shows that a geometrical property of a single arc can determine the type of the plane in which the arc can be embedded. Alternative proofs for this theorem were given by several authors see [ l , 14, 19, 20, In a paper due to appear see [21] Korchmaros has obtained a generalization of the above theorem for finite planes by weakening the hypothesis in the following way: the hexagons inscribed in K which are required to be pascalian are only those for which at least one of the lines joining two by two the three diagonal points is a chord or a tangent of the oval.

On abstract arcs of type m,n Abstract ovals were defined by Buekenhout [11] and their study led to many interesting problems of classification and existence. An example of an infinite abstract oval which is not embeddable in a projective plane has been given by Krier [22]. It is still an open question whether there exist or not finite. Barnabei and Zucchini [6] defined abstract arcs of type rn, n which generalize Buekenhout abstract ovals.

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Their definition includes as particular cases the k; n -arcs of types 0, rn, n , 0, n or m, n of a finite projective plane. The notion of abstract arc of type rn, n is founded on the definition of generalized involution of type m, n.

Let K be a finite set, whose elements we shall call points. A generalized involution of type m, n on K with 1 rn Results and problems in Galois geometry 3 An abstract arc of type m, n is given by a pair K ,9 , where K is a finite set and 9 is a set of generalized involutions of type m, n on K such that if "L" is the set of blocks of all involutions of 9, the following axioms hold: 1 two distinct points of K belong to exactly one block of "L"; 2 two blocks of "v; with empty intersection, belong to exactly one involution of 9; 3 in "L" there is at least one block of type n.

Clearly, k, n -arcs of types 0, m, n , 0, n or m, n of any finite projective plane can be seen as abstract arcs of type m, n if we consider as elements of I the partitions induced on the points of the arc by the lines of the pencils whose centers do not belong to the arc. Let K , 9 be any abstract arc of type m, n. The problem arises to decide whether or not the above structure can be completed in a suitable way to a projective plane. Barlotti 4 The knowledge of m r, q is important for applications to statistic and to coding theory, and a considerable amount of work has also been done to provide upper and lower bounds on m r, q for q 2 4 see [15,16,17,18,23].

In [24] is proved the existence of only two non-isomorphic caps in PG 4,3.

The packing problem arises and in general is far from being solved also for non desarguesian planes, for k-sets of kind s and for k, n -arcs and k, n -caps. References R. Artzy, Pascals theorem on an oval, Am. Monthly 75 Barlotti, Combinatorics of finite planes and other finite geometric structures, in: P.

Scherk ed.

Barnabei, Alcuni risultati sui k ; n -archi con tre caratteri, 1st. Lombard0 Accad.

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