Simultaneous systems of representatives for finite families of finite sets

Author:
Xing De Jia

Journal:
Proc. Amer. Math. Soc. **104** (1988), 33-36

MSC:
Primary 05A05; Secondary 11B99

MathSciNet review:
958037

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Abstract: Let and . It is proved that if and are two families of nonempty, pairwise disjoint sets such that and for all and , then the number of the sets such that is a minimal system of representatives for and is simultaneously a system of representatives for that satisfies , where with . This was conjectured by M. B. Nathanson [**3**] in 1985.

**[1]**Paul Erdős and Melvyn B. Nathanson,*Systems of distinct representatives and minimal bases in additive number theory*, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979) Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 89–107. MR**564925****[2]**Jia Xing-De,*On an open combinatorial problem of Erdös and Nathanson*, Chinese Ann. Math. (to appear).**[3]**Melvyn B. Nathanson,*Simultaneous systems of representatives for families of finite sets*, Proc. Amer. Math. Soc.**103**(1988), no. 4, 1322–1326. MR**955030**, 10.1090/S0002-9939-1988-0955030-2

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DOI:
http://dx.doi.org/10.1090/S0002-9939-1988-0958037-4

Article copyright:
© Copyright 1988
American Mathematical Society