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2-Bases of Quadruples

Published online by Cambridge University Press:  03 January 2006

ZOLTÁN FÜREDI  and
GYULA O. H. KATONA
ZOLTÁN FÜREDI
Affiliation:
Rényi Institute of Mathematics of the Hungarian Academy of Sciences, Budapest, PO Box 127, Hungary-1364 and Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL61801, USA (e-mail: furedi@renyi.hu, z-furedi@math.uiuc.edu)
GYULA O. H. KATONA
Affiliation:
Rényi Institute of Mathematics of the Hungarian Academy of Sciences, Budapest, PO Box 127, Hungary-1364 (e-mail: ohkatona@renyi.hu)
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Abstract

Let $\cal{B}(n, \leq 4)$ denote the subsets of $[n]:=\{ 1, 2, \dots, n\}$ of at most 4 elements. Suppose that $\cal{F}$ is a set system with the property that every member of $\cal{B}$ can be written as a union of (at most) two members of $\cal{F}$. (Such an $\cal{F}$ is called a 2-base of $\cal{B}$.) Here we answer a question of Erdős proving that \[|\FF|\geq 1+n+\binom{n}{2}- \Bigl\lfloor \frac{4}{3}n\Bigr\rfloor\], and this bound is best possible for $n\geq 8$.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 15 , Issue 1-2 , January 2006 , pp. 131 - 141
Copyright
2006 Cambridge University Press

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