Colloquia Mathematica Societatis
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چکیده
For any infinite cardinal K define a family .F of sets to be a Kalmost-disjoint family (K-ADF) iff U y has cardinality K, each member of .F has cardinality K, and the intersection of any two distinct members of F has cardinality strictly less than K . Define such a family to be a K-maximal almost-disjoint family (K-MADF) iff for every set S c U 'F of cardinality K there exists a set FEE JF* whose intersection with S has cardinality K . It is well-known (and easily seen) that if K has cofinality X < K, then any family of fewer than X disjoint sets each of cardinality K is a K-MADF while no family of cardinality X can be a KMADF. Thus for regular cardinals K there do not exist K-MADFs of cardinality K . In a private communication W . W i s t a r C o m f o r t asked if, however, for singular cardinals K there exist K-MADFs of cardinality K . We shall show that under certain conditions the answer is yes, but we do not know if the answer is ever no .
منابع مشابه
Colloquia Mathematica Societatis János Bolyai
1 . Our investigations might be suggested by a sort of optimalization problem which could be described vividly in practical terms as follows : suppose we are given a network of routes on which there are critical points endangered by fire. We want to place fire stations at an appropriate number of points from which any endangered point can be reached within a given time interval in case of emerg...
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