Following Erdos and Rado, three sets are said to form a "deltatriple" if any two of them have the same intersection. Let $F(n, 3)$ denote the largest cardinality of a family of subsets of an $n$--set not containing a delta--triple. It is not known whether $\limsup_{n \rightarrow \infty} n^{-1} \log F(n, 3)<1$. We say that a family of bipartitionsof an $n$--set is qualitatively 3/4-weakly 3--dependent if the common refinement of any 3 distinct partitions of the family has at least 6 non--empty classes, (i. e., at least 3/4 of the total.) Let $I(n)$ denote the maximum cardinality of such a family. We derive a simple relation between the exponential asymptotics of $F(n, 3)$ and $I(n)$ and show, as a consequence, that $\limsup_{n\rightarrow \infty} n^{-1} \log F(n,3)=1$ if and only if $\limsup_{n \rightarrow \infty}n^{-1} \log I(n)=1.$
Delta-systems and qualitative (in)dependence
2002
Abstract
Following Erdos and Rado, three sets are said to form a "deltatriple" if any two of them have the same intersection. Let $F(n, 3)$ denote the largest cardinality of a family of subsets of an $n$--set not containing a delta--triple. It is not known whether $\limsup_{n \rightarrow \infty} n^{-1} \log F(n, 3)<1$. We say that a family of bipartitionsof an $n$--set is qualitatively 3/4-weakly 3--dependent if the common refinement of any 3 distinct partitions of the family has at least 6 non--empty classes, (i. e., at least 3/4 of the total.) Let $I(n)$ denote the maximum cardinality of such a family. We derive a simple relation between the exponential asymptotics of $F(n, 3)$ and $I(n)$ and show, as a consequence, that $\limsup_{n\rightarrow \infty} n^{-1} \log F(n,3)=1$ if and only if $\limsup_{n \rightarrow \infty}n^{-1} \log I(n)=1.$I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
