Simultaneous systems of representatives for finite families of finite sets

Jia, Xing De

doi:10.1090/S0002-9939-1988-0958037-4

Simultaneous systems of representatives for finite families of finite sets
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Proc. Amer. Math. Soc. 104 (1988), 33-36 Request permission

Abstract:

Let $h \geq 2$ and $k \geq 1$. It is proved that if $\mathcal {S} = \{ {S_i}\} _{i = 1}^s$ and $\mathcal {T} = \{ {T_j}\} _{j = 1}^t$ are two families of nonempty, pairwise disjoint sets such that $|{S_i}| \leq h,|{T_j}| \leq k$ and ${S_i} \nsubseteq {T_j}$ for all $i$ and $j$, then the number $N(\mathcal {S},\mathcal {T})$ of the sets $X$ such that $X$ is a minimal system of representatives for $\mathcal {S}$ and $X$ is simultaneously a system of representatives for $\mathcal {T}$ that satisfies $N(\mathcal {S},\mathcal {T}) \leq {h^s}{(1 - (h - r)/{h^{q + 1}})^t}$, where $k = q(h - 1) + r$ with $0 \leq r \leq h - 2$. This was conjectured by M. B. Nathanson [3] in 1985.

References

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

On an open combinatorial problem of Erdös and Nathanson

Melvyn B. Nathanson, Simultaneous systems of representatives for families of finite sets, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1322–1326. MR 955030, DOI 10.1090/S0002-9939-1988-0955030-2