About two months ago, on the 4th of April, my friend, our friend, Giuseppe Tallini died in Rome where he was Professor of geometry at the University of 'La Sapienza'. Although he was su ering from lateral sclerosis, a terrible sickness indeed, until the end his mind was very clear and was directed to his family, to science, and to his students. Last September, in Montesilvano at the conference 'Combinatorics'94' he wanted to see the plans of the next three Conferences of the series ΓΏxed, the third one of which is to be held near Rome in the year 2000. I am sure that all his friends will do their best to grant his request! He was still planning in February to come here to Fort Collins and to present a paper. Unfortunately, this was not possible, but in spirit he is here with us.
In the ΓΏrst part of this talk, I wish to spend some time to remember the birth of the deep connection between R.C. Bose's school and the Italian school of Combinatorics (B. Segre, G. Zappa, L. Lombardo Radice, G. Tallini, etc.).
In 1954 B. Segre proved the result, conjectured by P. Kustaanheimo, that in a desarguesian projective plane of odd order, q, a (q + 1)-arc, i.e. a set of q + 1 points, no three of them collinear, is an irreducible conic. Following this ΓΏrst result, were several others concerning arcs and caps. It was natural that Prof. R.C. Bose, who was already using arcs and caps for certain applications, was deeply interested in these. So in 1963 Bose invited Beniamino Segre to spend one year as visiting Professor at his University: the University of North Carolina (at Chapel Hill). But since Segre was unable to pay a visit, the invitation was extended ΓΏrst to Tallini and then to me. In 1964 -1965, I went to Chapel Hill and this was one of the most wonderful years in my life. I had the opportunity to learn a little bit of English, and to visit several interesting places in North America. But the most important thing for me was that I got bound in a great friendship with R.C. Bose, a man of vivid geometric intuition, rich creative imagination and with a great culture. I also made friends with many of his associates and students.
Besides being interested in the quoted result of B. Segre, R.C. Bose was also attracted in particular to some papers of Tallini, at that time a young but very talented researcher at the University of Rome, who was working on di cult questions and had already found some very interesting results. I shall recall here a few examples of these.