Summary
When the kind invitation of Ron Graham and Jaroslav Nešetřil, to write in honour of Paul Erdős about aspects of his work, reached us, our first reaction was to follow it with great pleasure. Our second reaction was not as clear: Which one among the many subjects in mathematics, to which he has made fundamental contributions, should we choose?
Finally we just followed the most natural idea to write about an area which just had started to fascinate us: Density Theory for Integer Sequences.
More specifically we add here to the classical theory of primitive sequences and their sets of multiples results for cross-primitive sequences, a concept, which we introduce. We consider both, density properties for finite and infinite sequences. In the course of these investigations we naturally come across the main theorems in the classical theory and the predominance of results due to Paul Erdős becomes apparent. Several times he had exactly proved the theorems we wanted to prove! Many of them belong to his earliest contribution to mathematics in his early twenties.
Quite luckily our random approach led us to the perhaps most formidable period in Erdős’ work. It reminds us about a statement, which K. Reidemeister ([18], ch. 8) made about Carl Friedrich Gauss: “… Aber das Epochale ist doch die geniale Entdeckung des Jiinglings: die Zahlentheorie.“
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References
E. Sperner, “Ein Satz über Untermengen einer endlichen Menge”, Math. Z. 27, 544–548, 1928.
A.S. Besicovitch, “On the density of certain sequences of integers”, Math. Annal. 110, 336–341, 1934.
F. Behrend, “On sequences of numbers not divisible one by another”, J. London Math. Soc. 10, 42–44, 1935.
P. Erdős, “On primitive abundant numbers”, J. London Math. Soc. 10, 49–58, 1935.
P. Erdos, “Note on sequences of integers no one of which is divisible by any other”, J. Lond. Math. Soc. 10, 136–128, 1935.
P. Erdős, “Generalization of a theorem of Besicovitch”, J. London Math. Soc. 11, 92–98, 1936.
H. Davenport and P. Erdős, “On sequences of positive integers”, Acta Arithm. 2, 147–151, 1937.
S. Pillai, “On numbers which are not multiples of any other in the set”, Proc. Indian Acad. Sci. A 10, 392–394, 1939.
P. Erdős, “Integers with exactly k prime factors”, Ann. Math. II 49, 53–66, 1948.
F. Behrend, “Generalization of an inequality of Hulbrom and Rohrbach”, Bull. Ann. Math. Soc. 54, 681–684, 1948.
P. Erdős, “On the density of some sequences of integers”, Bull. Ann. Math. Soc 54, 685–692, 1948.
N.G. De Bruijn, C. van E. Tengbergen, and D. Kruyswijk, “On the set of divisors of a number”, Nieuw Arch. f. Wisk. Ser II, 23, 191–193, 1949–51.
H. Davenport and P. Erdos, “On sequences of positive integers”, J. Indian Math. Soc. 15, 19–24, 1951.
P. Erdős, Aufgabe 395 in Elem. Math. Basel 16, 21, 1961.
S.D. Chowla and T. Vijayaraghavan, On the largest prime divisors of numbers, J. of the Indian Math. Soc. 11, 31–37, 1947.
L.E. Dickson, “Finiteness of the odd perfect and primitive abundant numbers with n distinct prime factors”, American J. of Math. 35, 413–426, 1913.
H. Halberstam and K.F. Roth, “Sequences”, Oxford University Press, 1966, Springer-Verlag, New York, Heidelberg, Berlin 1983.
K. Reidemeister, “Raum und Zahl”, Springer-Verlag, Berlin, Göttingen, Heidelberg 1957.
R.R. Hall and G. Tenenbaum, “Divisors”, Cambridge Tracts in Mathematics 90, Cambridge University Press, Cambridge, New York, 1988.
P. Erdős, O. Sárközy, and E. Szemerédi, “On an extremal problem concerning primitive sequences”, J. London Math. Soc. 42, 484–488, 1967.
P. Erdős, A. Sárközy, and E. Szemerédi, “On a theorem of Behrend”, J. Australian Math. Soc. 7, 9–16, 1967.
P. Erdös, A. Sárközy, and E. Szemerédi, “On divisibility properties of sequences of integers”, Coll. Math. Soc. J. Bolyai 2, 35–49, 1970.
C. Pomerance and A. Sárközy, “On homogeneous multiplicative hybrid problems in number theory”, Acta Arith. 49, 291–302, 1988.
K. Yamamoto, “Logarithmic order of free distributive lattices”, J. Math. Soc. Japan 6, 343–353, 1954.
L.D. Meshalkin, “A generalization of Sperners theorem on the number of subsets of a finite set”, Theor. Probability Appl. 8, 203–204, 1963.
D. Lubell, “A short proof of Spemer’s theorem”, J. Combinatorial Theory 1, 299, 1966.
B. Bollobás, “On generalized graphs”, Acta Math. Acad. Sci. Hungar. 16, 447–452, 1965.
R. Ahlswede and Z. Zhang, “An identity in combinatorial extremal theory”, Adv. in Math., Vol. 80, No. 2, 137–151, 1990.
R. Ahlswede and N. Cai, “A generalization of the AZ-identity”, Combinatorica 13 (3), 241–247, 1993.
R. Ahlswede and Z. Zhang, “On cloud-antichains and related configurations”, Discrete Mathematics 85, 225–245, 1990.
R. Ahlswede and L.H. Khachatrian, “On extremal sets without coprimes”, Acta Arithmetica, LXVI 1, 89–99, 1994.
R. Ahlswede and L.H. Khachatrian, “Towards characterising equality in correlation inequalities”, Preprint 93–027, SFB 343 “Diskrete Strukturen in der Mathematik”, Universität Bielefeld, to appear in European J. of Combinatorics.
R. Ahlswede and L.H. Khachatrian, “Optimal pairs of incomparable clouds in Multisets”, Preprint 93, SFB 343 “Diskrete Strukturen in der Mathematik”, Universität Bielefeld, to appear in Graphs and Combinatorics.
R. Ahlswede and L.H. Khachatrian, “The maximal length of cloud-antichains”, Discrete Mathematics, Vol. 131, 9–15, 1994.
R. Ahlswede and L.H. Khachatrian, “Sharp bounds for cloud-antichains of length two”, Preprint 92–012, SFB 343 “Diskrete Strukturen in der Mathematik”, Universität Bielefeld.
H. Heilbronn, “On an inequality in the elementary theory of numbers”, Cambr. Phil. Soc. 33, 207–209, 1937.
H. Rohrbach, “Beweis einer zahlentheoretischen Ungleichung”, J. Reine u. Angew. Math. 177, 193–196, 1937.
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Ahlswede, R., Khachatrian, L.H. (1997). Classical Results on Primitive and Recent Results on Cross-Primitive Sequences. In: Graham, R.L., Nešetřil, J. (eds) The Mathematics of Paul Erdös I. Algorithms and Combinatorics, vol 13. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60408-9_9
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