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Intersection Theorems for Systems of Sets (II)
Author(s) -
Erdös P.,
Rado R.
Publication year - 1969
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s1-44.1.467
Subject(s) - intersection (aeronautics) , citation , computer science , library science , mathematics , information retrieval , engineering , aerospace engineering
In this paper we present the complete solution of the problem which was considered in [1], with the exception of the case in which both the given cardinal numbers are finite. The results of [1] will not be assumed. We begin by introducing some definitions, f A system Li = (Bv: veiV) of sets Bv, where v ranges over the index set N, is said to contain the system Eo = (A^ : n e M) if, for (j.o e M, the set A^o occurs in Sx at least as often as in I o , i.e. if |{v : v e JV; Bv = A,o}\ > |{/<: /< e M; A, = AM}| (n0 e M). If Sj contains Zo and, at the same time, So contains Zx then we do not distinguish between the systems So and 5^. The system Sx is called a (a, < b)-system if |N| = a and |5V| < & for veN. The system So is called a A(c)-system if |M| = c and i4Mi4^, = Atl2 Afl3 whenever ô» J"i. f*2> ^ 3 e M J Mo ̂ A*i; M2 ̂ M3The relation a A(b, c) (1)